Stochastic Calculus: Brownian Motion To SDEs Full Course Manual

This manual is the complete study source for the course. It keeps the core definitions, procedures, worked examples, practice tasks, and final-readiness checks in one place.

Course Spine

• Learner start state: single-variable and multivariable calculus basics, linear algebra for model notation, probability including conditional expectation, stochastic processes foundations or equivalent Markov/Poisson/Brownian exposure.
• Terminal capability: Introductory stochastic-calculus readiness for Brownian/Ito/SDE/Black-Scholes tasks and applied continuous-time-noise modeling.
• Key pillar points: brownian-filtration, quadratic-variation, ito-integral, ito-formula, sde-integral-form, geometric-brownian-motion, generator-feynman-kac, girsanov-change-of-measure, black-scholes-hedging, barrier-stopping, euler-maruyama, stochastic-model-choice.
• End evidence: A portfolio of Brownian, Ito-integral, Ito-formula, SDE, generator, Girsanov, Black-Scholes, stopping, simulation, and application-model-choice problems with derivations and assumption checks.

Chapter Map

Chapter 1: Brownian Motion, Filtration, And Adapted Processes

• Starts from: course start.
• Unlocks: Set up stochastic calculus with Brownian motion, current information, and adapted integrands.
• Ends when: learner can solve a representative problem and explain stochastic calculus starts by saying what noise is visible now and forbidding the integrand from seeing the future.
• Later dependency: Quadratic Variation And The Failure Of Ordinary Calculus.

Chapter 2: Quadratic Variation And The Failure Of Ordinary Calculus

• Starts from: Brownian Motion, Filtration, And Adapted Processes.
• Unlocks: Use quadratic variation to understand why Brownian calculus has an extra second-derivative term.
• Ends when: learner can solve a representative problem and explain brownian increments are small, but their squared increments add up to time rather than disappearing.
• Later dependency: Ito Integral And Ito Isometry.

Chapter 3: Ito Integral And Ito Isometry

• Starts from: Quadratic Variation And The Failure Of Ordinary Calculus.
• Unlocks: Define the Ito integral from adapted simple processes and compute its mean and variance using Ito isometry.
• Ends when: learner can solve a representative problem and explain the ito integral is a limit of sums that uses the information available before each noise increment arrives.
• Later dependency: Ito Formula, Product Rule, And Exponential Martingales.

Chapter 4: Ito Formula, Product Rule, And Exponential Martingales

• Starts from: Ito Integral And Ito Isometry.
• Unlocks: Apply Ito formula to transform stochastic processes and identify common martingales.
• Ends when: learner can solve a representative problem and explain ito formula is the chain rule with a volatility-curvature correction.
• Later dependency: SDEs, Drift, Diffusion, And Existence Intuition.

Chapter 5: SDEs, Drift, Diffusion, And Existence Intuition

• Starts from: Ito Formula, Product Rule, And Exponential Martingales.
• Unlocks: Read an SDE as a model with deterministic drift and random diffusion, and know the standard well-posedness warning signs.
• Ends when: learner can solve a representative problem and explain an sde says the next tiny change is a predictable tendency plus a noise shock scaled by the current state.
• Later dependency: Linear SDEs, Geometric Brownian Motion, And OU Models.

Chapter 6: Linear SDEs, Geometric Brownian Motion, And OU Models

• Starts from: SDEs, Drift, Diffusion, And Existence Intuition.
• Unlocks: Solve and interpret common SDE families such as geometric Brownian motion and Ornstein-Uhlenbeck mean reversion.
• Ends when: learner can solve a representative problem and explain some sdes have structure strong enough to solve explicitly after the right transform or integrating factor.
• Later dependency: Generators, Kolmogorov Equations, And Feynman-Kac.

Chapter 7: Generators, Kolmogorov Equations, And Feynman-Kac

• Starts from: Linear SDEs, Geometric Brownian Motion, And OU Models.
• Unlocks: Connect SDEs to partial differential equations through generators and expectation representations.
• Ends when: learner can solve a representative problem and explain the generator tells how expected future values start changing from the current state.
• Later dependency: Change Of Measure And Girsanov.

Chapter 8: Change Of Measure And Girsanov

• Starts from: Generators, Kolmogorov Equations, And Feynman-Kac.
• Unlocks: Understand how changing probability measures can turn drift-adjusted Brownian motion into Brownian motion under a new measure.
• Ends when: learner can solve a representative problem and explain girsanov changes the likelihood of paths so that what looked like drift under one measure can become part of brownian motion under another.
• Later dependency: Black-Scholes Pricing And Hedging.

Chapter 9: Black-Scholes Pricing And Hedging

• Starts from: Change Of Measure And Girsanov.
• Unlocks: Derive the Black-Scholes PDE and explain option price as a hedged replication or risk-neutral expectation.
• Ends when: learner can solve a representative problem and explain a continuously adjusted hedge can cancel brownian risk and force the remaining position to earn the risk-free rate.
• Later dependency: Stopping, Barriers, And Continuous-Time Martingales.

Chapter 10: Stopping, Barriers, And Continuous-Time Martingales

• Starts from: Black-Scholes Pricing And Hedging.
• Unlocks: Use continuous-time martingales and stopping arguments carefully for hitting and barrier problems.
• Ends when: learner can solve a representative problem and explain stopping tools are powerful because martingales keep fair value, but only when the stopping conditions are legal.
• Later dependency: Euler-Maruyama Simulation And Numerical Error.

Chapter 11: Euler-Maruyama Simulation And Numerical Error

• Starts from: Stopping, Barriers, And Continuous-Time Martingales.
• Unlocks: Simulate SDE paths with Euler-Maruyama and explain discretization, sampling error, and diagnostic checks.
• Ends when: learner can solve a representative problem and explain simulation replaces infinitesimal drift and noise with small time steps; the model becomes useful only when the error story is honest.
• Later dependency: Model Choice, Applications, And Readiness Synthesis.

Chapter 12: Model Choice, Applications, And Readiness Synthesis

• Starts from: Euler-Maruyama Simulation And Numerical Error.
• Unlocks: Choose stochastic-calculus tools for applied problems and defend the assumptions, calculations, and limitations.
• Ends when: learner can solve a representative problem and explain stochastic calculus is useful when the application has continuous-time noise, but the model only earns trust when assumptions and outputs are checked.
• Later dependency: final readiness synthesis.

Detailed Knowledge Manual

The learner must know the definitions, assumptions, notation, procedures, examples, traps, repair moves, and model-choice cues in every chapter. A formula is not considered learned until the learner can state the stochastic object, justify the assumptions, carry out a representative calculation, and interpret the answer in ordinary language.

Evaluation

Evaluation uses formative checks, exercise families, worked solutions, delayed review, transfer tasks, derivation or counterexample prompts, and a timed mock final. The course is exam-prepared only when the learner can solve representative original tasks and repair misses without hidden hints.

Expanded Study Manual Layer

Expanded Chapter 1: Brownian Motion, Filtration, And Adapted Processes

Core story. Stochastic calculus starts by saying what noise is visible now and forbidding the integrand from seeing the future. In applications such as asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. Brownian motion \(B_t\) is adapted to a filtration \(\mathcal{F}_t\) , and a process \(H_t\) is adapted when \(H_t\) is \(\mathcal{F}_t\) -measurable for each t. The formula or test to keep visible is \(B_t-B_s\) is independent of \(\mathcal{F}_s\) , and \(B_t-B_s\sim\mathcal{N}(0,t-s)\) for \(s<t\) under the Brownian filtration. The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Name the probability space, filtration, Brownian driver, adapted process, time interval, and integrability condition before writing an integral. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations. Define the state and random mechanism, state the assumption that makes the model legal, use \(B_t-B_s\) is independent of \(\mathcal{F}_s\) , and \(B_t-B_s\sim\mathcal{N}(0,t-s)\) for \(s<t\) under the Brownian filtration, and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is letting a trading strategy or integrand depend on tomorrow Brownian increment. The repair is: ask what is known at time t and reject any rule that uses future noise The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Decide whether \(H_t=B_{t/2}\) , \(H_t=B_{t+1}\) , and \(H_t=\mathbf{1}_{\{B_t>0\}}\) are adapted to the Brownian filtration. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose letting a trading strategy or integrand depend on tomorrow Brownian increment. Transfer: Explain why a self-financing trading strategy must be adapted before a pricing model can trust it. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Brownian Motion, Filtration, And Adapted Processes can support asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 2: Quadratic Variation And The Failure Of Ordinary Calculus

Core story. Brownian increments are small, but their squared increments add up to time rather than disappearing. In applications such as realized volatility, rough price paths, diffusion measurement, signal noise accumulation, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. Along refining partitions, \(\sum_k(B_{t_{k+1}}-B_{t_k})^2\) converges in probability to t, while total variation is infinite. The formula or test to keep visible is \((dB_t)^2=dt\) , \(dt\,dB_t=0\) , and \((dt)^2=0\) in Itô differential bookkeeping. The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Compare orders of magnitude: dB is size \(\sqrt{dt}\) , so \((dB)^2\) contributes like dt and \(dB\,dt\) or \((dt)^2\) disappear. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with realized volatility, rough price paths, diffusion measurement, signal noise accumulation. Define the state and random mechanism, state the assumption that makes the model legal, use \((dB_t)^2=dt\) , \(dt\,dB_t=0\) , and \((dt)^2=0\) in Itô differential bookkeeping, and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is using ordinary chain rule and missing the quadratic variation correction. The repair is: write the Ito table before differentiating any function of Brownian motion The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Explain why \(d(B_t^2)\) is not just \(2B_t\,dB_t\) and identify the missing term. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose using ordinary chain rule and missing the quadratic variation correction. Transfer: Connect quadratic variation to realized volatility from high-frequency returns. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Quadratic Variation And The Failure Of Ordinary Calculus can support realized volatility, rough price paths, diffusion measurement, signal noise accumulation. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 3: Ito Integral And Ito Isometry

Core story. The Ito integral is a limit of sums that uses the information available before each noise increment arrives. In applications such as noise-driven accumulated exposure, hedging error variance, stochastic signal filtering, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. For square-integrable adapted H, \(\int_0^TH_t\,dB_t\) is the L2 limit of left-endpoint sums, with \(\mathbb{E}[(\int H\,dB)^2]=\mathbb{E}[\int H^2\,dt]\) . The formula or test to keep visible is \(\mathbb{E}\!\left[\int_0^T H_t\,dB_t\right]=0\) and \(\mathbb{E}\!\left[\left(\int_0^T H_t\,dB_t\right)^2\right]=\mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]\) . The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Approximate by simple adapted steps, use independent Brownian increments for mean zero, apply Ito isometry for second moments, then pass to limits. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with noise-driven accumulated exposure, hedging error variance, stochastic signal filtering. Define the state and random mechanism, state the assumption that makes the model legal, use \(\mathbb{E}\!\left[\int_0^T H_t\,dB_t\right]=0\) and \(\mathbb{E}\!\left[\left(\int_0^T H_t\,dB_t\right)^2\right]=\mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]\) , and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is treating dB like ordinary dt and pulling random future terms out of the integral. The repair is: check adaptedness and square integrability before applying the isometry The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Compute \(\mathbb{E}\!\left[\left(\int_0^TB_t\,dB_t\right)^2\right]\) using Ito isometry. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose treating dB like ordinary dt and pulling random future terms out of the integral. Transfer: Compute the variance of accumulated noise exposure when \(H_t\) is a deterministic seasonal load. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Ito Integral And Ito Isometry can support noise-driven accumulated exposure, hedging error variance, stochastic signal filtering. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 4: Ito Formula, Product Rule, And Exponential Martingales

Core story. Ito formula is the chain rule with a volatility-curvature correction. In applications such as log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. If \(dX_t=\mu_t\,dt+\sigma_t\,dB_t\) , then \(df(t,X_t)=(f_t+\mu f_x+\tfrac12\sigma^2f_{xx})\,dt+\sigma f_x\,dB_t\) . The formula or test to keep visible is \(d(fg)=f\,dg+g\,df+d[f,g]\) , and \(\exp(\theta B_t-\tfrac12\theta^2t)\) is a martingale. The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Identify drift, diffusion, derivatives of f, insert into Ito formula, collect dt and dB terms, and interpret martingale versus drift parts. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance. Define the state and random mechanism, state the assumption that makes the model legal, use \(d(fg)=f\,dg+g\,df+d[f,g]\) , and \(\exp(\theta B_t-\tfrac12\theta^2t)\) is a martingale, and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is forgetting that product rules include a quadratic covariation term. The repair is: compute the dX dY term before simplifying a product or chain rule The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Use Ito formula to find \(d(\log S_t)\) when \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) . Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose forgetting that product rules include a quadratic covariation term. Transfer: Derive the dynamics of \(\exp(B_t-t/2)\) and explain why it has zero drift. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Ito Formula, Product Rule, And Exponential Martingales can support log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 5: SDEs, Drift, Diffusion, And Existence Intuition

Core story. An SDE says the next tiny change is a predictable tendency plus a noise shock scaled by the current state. In applications such as population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. An Ito SDE has form \(dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\) with adapted solution satisfying the integral equation. The formula or test to keep visible is \(X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\) . The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. State the integral form, identify drift and diffusion, check initial condition, note regularity assumptions, and explain the modeling meaning of each coefficient. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials. Define the state and random mechanism, state the assumption that makes the model legal, use \(X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\) , and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is treating an SDE as an ordinary differential equation with a random term pasted on after solving. The repair is: write the integral equation and keep the stochastic integral inside the solution definition The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: For \(dX_t=aX_t\,dt+bX_t\,dB_t\) , identify drift, diffusion, and why the solution is not \(e^{(a+b)t}\) . Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose treating an SDE as an ordinary differential equation with a random term pasted on after solving. Transfer: Choose drift and diffusion terms for a noisy mean-reverting temperature model. Teach-back: explain why the model fits and where it could fail. Application note. In real work, SDEs, Drift, Diffusion, And Existence Intuition can support population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 6: Linear SDEs, Geometric Brownian Motion, And OU Models

Core story. Some SDEs have structure strong enough to solve explicitly after the right transform or integrating factor. In applications such as stock prices, interest-rate factors, commodity spreads, physical velocity with friction, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. Geometric Brownian motion solves \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) with \(S_t=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\) . The formula or test to keep visible is OU: \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) , with mean \(\theta+(X_0-\theta)e^{-\kappa t}\) . The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Try log transform for multiplicative noise, integrating factor for linear SDEs, and mean/variance formulas for Gaussian mean-reverting models. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with stock prices, interest-rate factors, commodity spreads, physical velocity with friction. Define the state and random mechanism, state the assumption that makes the model legal, use OU: \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) , with mean \(\theta+(X_0-\theta)e^{-\kappa t}\) , and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is forgetting the \(-\tfrac12\sigma^2\) correction in geometric Brownian motion log dynamics. The repair is: derive log \(S_t\) with Ito formula rather than guessing the exponential solution The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Solve geometric Brownian motion and compute E[\(S_t\) ]. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose forgetting the \(-\tfrac12\sigma^2\) correction in geometric Brownian motion log dynamics. Transfer: Use an OU model to explain mean-reverting short-rate behavior. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Linear SDEs, Geometric Brownian Motion, And OU Models can support stock prices, interest-rate factors, commodity spreads, physical velocity with friction. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 7: Generators, Kolmogorov Equations, And Feynman-Kac

Core story. The generator tells how expected future values start changing from the current state. In applications such as option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. For \(dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t\) , generator \(\mathcal{L}f=bf_x+\tfrac12\sigma^2f_{xx}\) ; Feynman-Kac links parabolic PDEs to discounted expectations. The formula or test to keep visible is \(u_t+\mathcal{L}u-ru+g=0\) with terminal condition \(u(T,x)=\phi(x)\) has an expectation representation under standard assumptions. The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Compute L, choose the terminal or boundary payoff, write the backward equation, and interpret the solution as an expectation under the diffusion. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews. Define the state and random mechanism, state the assumption that makes the model legal, use \(u_t+\mathcal{L}u-ru+g=0\) with terminal condition \(u(T,x)=\phi(x)\) has an expectation representation under standard assumptions, and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is mixing forward density equations with backward value equations. The repair is: ask whether the unknown is a distribution moving forward or a value function looking backward from payoff The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Find the generator of geometric Brownian motion and write the backward pricing PDE for a terminal payoff. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose mixing forward density equations with backward value equations. Transfer: Use a generator to approximate a diffusion hitting probability by solving a boundary value equation. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Generators, Kolmogorov Equations, And Feynman-Kac can support option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 8: Change Of Measure And Girsanov

Core story. Girsanov changes the likelihood of paths so that what looked like drift under one measure can become part of Brownian motion under another. In applications such as risk-neutral pricing, likelihood ratios, filtering, rare-event simulation, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. Under suitable Novikov-type conditions, an exponential martingale defines a new measure under which \(B_t\) minus an integrated drift becomes Brownian motion. The formula or test to keep visible is \(Z_t=\exp\!\left(-\int\theta\,dB-\tfrac12\int\theta^2\,dt\right)\) , and \(W_t=B_t+\int\theta\,dt\) is Brownian under the new measure. The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Identify market price of risk or drift shift, build the exponential density, state the condition, transform Brownian motion, and rewrite the SDE under the new measure. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with risk-neutral pricing, likelihood ratios, filtering, rare-event simulation. Define the state and random mechanism, state the assumption that makes the model legal, use \(Z_t=\exp\!\left(-\int\theta\,dB-\tfrac12\int\theta^2\,dt\right)\) , and \(W_t=B_t+\int\theta\,dt\) is Brownian under the new measure, and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is changing drift without changing the probability measure or density process. The repair is: write the density process first and state which measure each Brownian motion belongs to The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Given \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) , choose \(\theta\) so discounted S has drift zero under the new measure. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose changing drift without changing the probability measure or density process. Transfer: Explain why risk-neutral pricing changes expected growth but not the volatility coefficient. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Change Of Measure And Girsanov can support risk-neutral pricing, likelihood ratios, filtering, rare-event simulation. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 9: Black-Scholes Pricing And Hedging

Core story. A continuously adjusted hedge can cancel Brownian risk and force the remaining position to earn the risk-free rate. In applications such as equity options, hedging desks, implied volatility, risk management, model-risk analysis, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. Under geometric Brownian stock dynamics with constant r and \(\sigma\) , option value V satisfies \(V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0\) with terminal payoff. The formula or test to keep visible is The call price is \(C=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)\) under the standard Black-Scholes assumptions. The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Apply Ito to V(t,\(S_t\) ), form delta hedge, cancel dB term, impose risk-free return, state terminal condition, and connect to risk-neutral expectation. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with equity options, hedging desks, implied volatility, risk management, model-risk analysis. Define the state and random mechanism, state the assumption that makes the model legal, use The call price is \(C=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)\) under the standard Black-Scholes assumptions, and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is memorizing the formula without knowing the assumptions and hedge argument. The repair is: derive the PDE before using the closed-form call expression The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Derive the Black-Scholes PDE for a European option from Ito formula and a delta hedge. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose memorizing the formula without knowing the assumptions and hedge argument. Transfer: Explain how higher volatility changes call option value through convex payoff exposure. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Black-Scholes Pricing And Hedging can support equity options, hedging desks, implied volatility, risk management, model-risk analysis. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 10: Stopping, Barriers, And Continuous-Time Martingales

Core story. Stopping tools are powerful because martingales keep fair value, but only when the stopping conditions are legal. In applications such as barrier options, ruin probabilities, hitting thresholds, sequential detection, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. For suitable stopping times tau and martingales \(M_t\) , optional sampling gives \(\mathbb{E}[M_\tau]=\mathbb{E}[M_0]\) under boundedness or integrability hypotheses. The formula or test to keep visible is \(B_t^2-t\) is a martingale; \(\exp(\theta B_t-\tfrac12\theta^2t)\) is also a martingale. The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Build a martingale from Brownian motion or an SDE, identify the stopping time, localize or bound it, apply optional sampling, and solve boundary equations. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with barrier options, ruin probabilities, hitting thresholds, sequential detection. Define the state and random mechanism, state the assumption that makes the model legal, use \(B_t^2-t\) is a martingale; \(\exp(\theta B_t-\tfrac12\theta^2t)\) is also a martingale, and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is using optional sampling at an infinite-mean hitting time without justification. The repair is: truncate the stopping time or state the theorem conditions before passing to a limit The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Use a Brownian martingale to compute the probability of hitting upper barrier a before lower barrier -b. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose using optional sampling at an infinite-mean hitting time without justification. Transfer: Price or bound a simple barrier-style payoff by identifying the relevant hitting event. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Stopping, Barriers, And Continuous-Time Martingales can support barrier options, ruin probabilities, hitting thresholds, sequential detection. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 11: Euler-Maruyama Simulation And Numerical Error

Core story. Simulation replaces infinitesimal drift and noise with small time steps; the model becomes useful only when the error story is honest. In applications such as risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. Euler-Maruyama uses \(X_{n+1}=X_n+b(t_n,X_n)\Delta t+\sigma(t_n,X_n)\Delta B_n\) with \(\Delta B_n\sim\mathcal{N}(0,\Delta t)\) . The formula or test to keep visible is \(\Delta B_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0,1)\) . The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Choose time step, generate Brownian increments, update drift and diffusion, repeat many paths, estimate quantity, and separate Monte Carlo error from time-discretization error. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation. Define the state and random mechanism, state the assumption that makes the model legal, use \(\Delta B_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0,1)\) , and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is using too-large time steps and mistaking discretization bias for model behavior. The repair is: halve the time step, increase path count, and compare stable quantities The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Write the Euler-Maruyama update for \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) and describe one convergence check. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose using too-large time steps and mistaking discretization bias for model behavior. Transfer: Simulate a geometric Brownian portfolio loss distribution and report uncertainty in the estimate. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Euler-Maruyama Simulation And Numerical Error can support risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Chapter 12: Model Choice, Applications, And Readiness Synthesis

Core story. Stochastic calculus is useful when the application has continuous-time noise, but the model only earns trust when assumptions and outputs are checked. In applications such as option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations, this idea prevents a learner from treating uncertainty as vague noise. It forces the learner to specify the random object, information, state, time index, and target quantity. Formal payload. A stochastic-calculus model specifies filtration, driving noise, SDE coefficients, payoff or observable, measure, numerical method, and validation target. The formula or test to keep visible is Model choice checklist: information, state, drift, diffusion, boundary, payoff, measure, numerical scheme, validation. The learner should be able to derive or justify this statement at the level appropriate to the course, not only quote it. Procedure payload. Classify the problem, choose Brownian/SDE/tool family, state assumptions, compute or simulate, interpret units, run a trap check, and state model risk. In a timed exam, the first line should identify the model and assumptions; the second line should state the formula or equation; the last line should interpret the result. Worked example model. Start with option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations. Define the state and random mechanism, state the assumption that makes the model legal, use Model choice checklist: information, state, drift, diffusion, boundary, payoff, measure, numerical scheme, validation, and write the result as a sentence. If a numerical value is requested, keep the symbolic setup first so a grader can see the method. Classic trap and repair. The common wrong move is using Black-Scholes or GBM as a universal default instead of matching data and assumptions. The repair is: force a model-choice justification before any formula and list at least one failure mode The learner should practice recognizing the trap from a wrong worked solution, because human exams often reward diagnosis and not just clean computation. Exercise questions. Recall: state the definition. Guided: Choose between GBM, OU, reflected diffusion, or a jump model for four short scenarios and justify one limitation of each choice. Independent: change the parameters and solve. Variation: solve the same structure in a different application. Trap repair: diagnose using Black-Scholes or GBM as a universal default instead of matching data and assumptions. Transfer: Design a small stochastic-calculus modeling memo for a hedging, risk, or noisy-physics application. Teach-back: explain why the model fits and where it could fail. Application note. In real work, Model Choice, Applications, And Readiness Synthesis can support option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations. The manual asks for model risk because a correct computation under wrong assumptions is still a bad solution.

Expanded Worked Example And Exercise Bank

Exercise Bank 1: Brownian Motion, Filtration, And Adapted Processes

1. Definition drill: write the formal statement and identify each symbol in \(B_t-B_s\) is independent of \(\mathcal{F}_s\) , and \(B_t-B_s\sim\mathcal{N}(0,t-s)\) for \(s<t\) under the Brownian filtration.
2. Computation drill: Decide whether \(H_t=B_{t/2}\) , \(H_t=B_{t+1}\) , and \(H_t=\mathbf{1}_{\{B_t>0\}}\) are adapted to the Brownian filtration.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations.
4. Trap drill: repair a solution that makes the error letting a trading strategy or integrand depend on tomorrow Brownian increment.
5. Transfer drill: Explain why a self-financing trading strategy must be adapted before a pricing model can trust it.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: \(B_{t/2}\) and \(\mathbf{1}_{\{B_t>0\}}\) are adapted; \(B_{t+1}\) is not because it uses future information. The final answer must include assumptions and interpretation.

Exercise Bank 2: Quadratic Variation And The Failure Of Ordinary Calculus

1. Definition drill: write the formal statement and identify each symbol in \((dB_t)^2=dt\) , \(dt\,dB_t=0\) , and \((dt)^2=0\) in Itô differential bookkeeping.
2. Computation drill: Explain why \(d(B_t^2)\) is not just \(2B_t\,dB_t\) and identify the missing term.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in realized volatility, rough price paths, diffusion measurement, signal noise accumulation.
4. Trap drill: repair a solution that makes the error using ordinary chain rule and missing the quadratic variation correction.
5. Transfer drill: Connect quadratic variation to realized volatility from high-frequency returns.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: Ito formula gives \(d(B_t^2)=2B_t\,dB_t+dt\) because Brownian quadratic variation contributes dt. The final answer must include assumptions and interpretation.

Exercise Bank 3: Ito Integral And Ito Isometry

1. Definition drill: write the formal statement and identify each symbol in \(\mathbb{E}\!\left[\int_0^T H_t\,dB_t\right]=0\) and \(\mathbb{E}\!\left[\left(\int_0^T H_t\,dB_t\right)^2\right]=\mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]\) .
2. Computation drill: Compute \(\mathbb{E}\!\left[\left(\int_0^TB_t\,dB_t\right)^2\right]\) using Ito isometry.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in noise-driven accumulated exposure, hedging error variance, stochastic signal filtering.
4. Trap drill: repair a solution that makes the error treating dB like ordinary dt and pulling random future terms out of the integral.
5. Transfer drill: Compute the variance of accumulated noise exposure when \(H_t\) is a deterministic seasonal load.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: Use \(\mathbb{E}\!\left[\int_0^TB_t^2\,dt\right]=\int_0^T\mathbb{E}[B_t^2]\,dt=\int_0^Tt\,dt=\dfrac{T^2}{2}\) . The final answer must include assumptions and interpretation.

Exercise Bank 4: Ito Formula, Product Rule, And Exponential Martingales

1. Definition drill: write the formal statement and identify each symbol in \(d(fg)=f\,dg+g\,df+d[f,g]\) , and \(\exp(\theta B_t-\tfrac12\theta^2t)\) is a martingale.
2. Computation drill: Use Ito formula to find \(d(\log S_t)\) when \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) .
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance.
4. Trap drill: repair a solution that makes the error forgetting that product rules include a quadratic covariation term.
5. Transfer drill: Derive the dynamics of \(\exp(B_t-t/2)\) and explain why it has zero drift.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: \(d\log S_t=(\mu-\tfrac12\sigma^2)\,dt+\sigma\,dB_t\) . The final answer must include assumptions and interpretation.

Exercise Bank 5: SDEs, Drift, Diffusion, And Existence Intuition

1. Definition drill: write the formal statement and identify each symbol in \(X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\) .
2. Computation drill: For \(dX_t=aX_t\,dt+bX_t\,dB_t\) , identify drift, diffusion, and why the solution is not \(e^{(a+b)t}\) .
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials.
4. Trap drill: repair a solution that makes the error treating an SDE as an ordinary differential equation with a random term pasted on after solving.
5. Transfer drill: Choose drift and diffusion terms for a noisy mean-reverting temperature model.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: Drift is aX_t, diffusion is bX_t, and Ito correction appears in the exponential solution. The final answer must include assumptions and interpretation.

Exercise Bank 6: Linear SDEs, Geometric Brownian Motion, And OU Models

1. Definition drill: write the formal statement and identify each symbol in OU: \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) , with mean \(\theta+(X_0-\theta)e^{-\kappa t}\) .
2. Computation drill: Solve geometric Brownian motion and compute E[\(S_t\) ].
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in stock prices, interest-rate factors, commodity spreads, physical velocity with friction.
4. Trap drill: repair a solution that makes the error forgetting the \(-\tfrac12\sigma^2\) correction in geometric Brownian motion log dynamics.
5. Transfer drill: Use an OU model to explain mean-reverting short-rate behavior.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: \(S_t=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\) and \(\mathbb{E}[S_t]=S_0e^{\mu t}\) . The final answer must include assumptions and interpretation.

Exercise Bank 7: Generators, Kolmogorov Equations, And Feynman-Kac

1. Definition drill: write the formal statement and identify each symbol in \(u_t+\mathcal{L}u-ru+g=0\) with terminal condition \(u(T,x)=\phi(x)\) has an expectation representation under standard assumptions.
2. Computation drill: Find the generator of geometric Brownian motion and write the backward pricing PDE for a terminal payoff.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews.
4. Trap drill: repair a solution that makes the error mixing forward density equations with backward value equations.
5. Transfer drill: Use a generator to approximate a diffusion hitting probability by solving a boundary value equation.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: \(\mathcal{L}f=\mu xf_x+\tfrac12\sigma^2x^2f_{xx}\) ; the value solves \(u_t+\mathcal{L}u-ru=0\) with terminal payoff. The final answer must include assumptions and interpretation.

Exercise Bank 8: Change Of Measure And Girsanov

1. Definition drill: write the formal statement and identify each symbol in \(Z_t=\exp\!\left(-\int\theta\,dB-\tfrac12\int\theta^2\,dt\right)\) , and \(W_t=B_t+\int\theta\,dt\) is Brownian under the new measure.
2. Computation drill: Given \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) , choose \(\theta\) so discounted S has drift zero under the new measure.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in risk-neutral pricing, likelihood ratios, filtering, rare-event simulation.
4. Trap drill: repair a solution that makes the error changing drift without changing the probability measure or density process.
5. Transfer drill: Explain why risk-neutral pricing changes expected growth but not the volatility coefficient.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: Use \(\theta=(\mu-r)/\sigma\) with the correct sign convention and rewrite \(dS_t=rS_t\,dt+\sigma S_t\,dW_t\) under Q. The final answer must include assumptions and interpretation.

Exercise Bank 9: Black-Scholes Pricing And Hedging

1. Definition drill: write the formal statement and identify each symbol in The call price is \(C=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)\) under the standard Black-Scholes assumptions.
2. Computation drill: Derive the Black-Scholes PDE for a European option from Ito formula and a delta hedge.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in equity options, hedging desks, implied volatility, risk management, model-risk analysis.
4. Trap drill: repair a solution that makes the error memorizing the formula without knowing the assumptions and hedge argument.
5. Transfer drill: Explain how higher volatility changes call option value through convex payoff exposure.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: Compute dV, hold \(\Delta=V_S\) shares to cancel dB, and require the riskless residual to grow at r. The final answer must include assumptions and interpretation.

Exercise Bank 10: Stopping, Barriers, And Continuous-Time Martingales

1. Definition drill: write the formal statement and identify each symbol in \(B_t^2-t\) is a martingale; \(\exp(\theta B_t-\tfrac12\theta^2t)\) is also a martingale.
2. Computation drill: Use a Brownian martingale to compute the probability of hitting upper barrier a before lower barrier -b.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in barrier options, ruin probabilities, hitting thresholds, sequential detection.
4. Trap drill: repair a solution that makes the error using optional sampling at an infinite-mean hitting time without justification.
5. Transfer drill: Price or bound a simple barrier-style payoff by identifying the relevant hitting event.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: Stop \(B_t\) at the bounded hitting time of the interval endpoints and solve \(0=ap-b(1-p)\) . The final answer must include assumptions and interpretation.

Exercise Bank 11: Euler-Maruyama Simulation And Numerical Error

1. Definition drill: write the formal statement and identify each symbol in \(\Delta B_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0,1)\) .
2. Computation drill: Write the Euler-Maruyama update for \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) and describe one convergence check.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation.
4. Trap drill: repair a solution that makes the error using too-large time steps and mistaking discretization bias for model behavior.
5. Transfer drill: Simulate a geometric Brownian portfolio loss distribution and report uncertainty in the estimate.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: \(X_{n+1}=X_n+\kappa(\theta-X_n)\Delta t+\sigma\sqrt{\Delta t}Z_n\) ; compare estimates across smaller Delta t. The final answer must include assumptions and interpretation.

Exercise Bank 12: Model Choice, Applications, And Readiness Synthesis

1. Definition drill: write the formal statement and identify each symbol in Model choice checklist: information, state, drift, diffusion, boundary, payoff, measure, numerical scheme, validation.
2. Computation drill: Choose between GBM, OU, reflected diffusion, or a jump model for four short scenarios and justify one limitation of each choice.
3. Interpretation drill: after solving, write one sentence for a nontechnical stakeholder in option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations.
4. Trap drill: repair a solution that makes the error using Black-Scholes or GBM as a universal default instead of matching data and assumptions.
5. Transfer drill: Design a small stochastic-calculus modeling memo for a hedging, risk, or noisy-physics application.
6. Teach-back drill: explain why this is not merely a deterministic calculation with random-looking inputs. Worked-solution sketch: Match multiplicative growth to GBM, mean reversion to OU, boundaries to reflected or killed diffusions, and discontinuous shocks to jump models outside this course boundary. The final answer must include assumptions and interpretation.

Final Exam Readiness

Final readiness means the learner can solve closed-note, time-limited stochastic-calculus tasks across Brownian motion, filtrations, quadratic variation, Ito integrals, Ito formula, SDEs, geometric Brownian motion, Ornstein-Uhlenbeck models, generators, Feynman-Kac, Girsanov, Black-Scholes, stopping, and Euler-Maruyama.

Real Applications Lab

• Black-Scholes option pricing: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• delta hedging: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• risk-neutral valuation: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• interest-rate mean reversion: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• realized volatility: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• barrier risk: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• Monte Carlo risk simulation: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• diffusion physics: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• stochastic climate perturbations: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.
• noisy population models: build the stochastic state, choose the time index, state assumptions, calculate one target, and name one limitation.

Original Timed Mock Final Blueprint

Work closed-note. Suggested duration: 150 minutes. Use original mixed-practice tasks for the course scope. For every answer, show model, assumptions, calculation, interpretation, and limitation.

1. adaptedness and filtration checks: solve one original problem and explain why the selected model is appropriate.
2. quadratic variation explanation: solve one original problem and explain why the selected model is appropriate.
3. Ito isometry moment calculation: solve one original problem and explain why the selected model is appropriate.
4. Ito formula derivation: solve one original problem and explain why the selected model is appropriate.
5. SDE integral-form reading: solve one original problem and explain why the selected model is appropriate.
6. geometric Brownian motion and OU solution: solve one original problem and explain why the selected model is appropriate.
7. generator and Feynman-Kac PDE setup: solve one original problem and explain why the selected model is appropriate.
8. Girsanov drift-shift calculation: solve one original problem and explain why the selected model is appropriate.
9. Black-Scholes PDE and hedge derivation: solve one original problem and explain why the selected model is appropriate.
10. barrier hitting and optional sampling: solve one original problem and explain why the selected model is appropriate.
11. Euler-Maruyama simulation update: solve one original problem and explain why the selected model is appropriate.
12. application model-choice memo: solve one original problem and explain why the selected model is appropriate.

Practice, Solutions, And Review

The sections below teach, practice, solve, review, and rehearse the course knowledge.

Study Chapters

These chapters collect definitions, procedures, worked examples, traps, applications, and exercises for independent study.

Chapter 1: Brownian Motion, Filtration, And Adapted Processes

Why it matters: asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations is hard to reason about without a model for random evolution. Stochastic calculus starts by saying what noise is visible now and forbidding the integrand from seeing the future. Definition and statement: Brownian motion \(B_t\) is adapted to a filtration \(\mathcal{F}_t\) , and a process \(H_t\) is adapted when \(H_t\) is \(\mathcal{F}_t\) -measurable for each t. Procedure: Name the probability space, filtration, Brownian driver, adapted process, time interval, and integrability condition before writing an integral. Main formula or test: \(B_t-B_s\) is independent of \(\mathcal{F}_s\) , and \(B_t-B_s\sim\mathcal{N}(0,t-s)\) for \(s<t\) under the Brownian filtration. Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \(B_t-B_s\) is independent of \(\mathcal{F}_s\) , and \(B_t-B_s\sim\mathcal{N}(0,t-s)\) for \(s<t\) under the Brownian filtration.
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations. Classic trap: letting a trading strategy or integrand depend on tomorrow Brownian increment Repair: ask what is known at time t and reject any rule that uses future noise Guided practice: Decide whether \(H_t=B_{t/2}\) , \(H_t=B_{t+1}\) , and \(H_t=\mathbf{1}_{\{B_t>0\}}\) are adapted to the Brownian filtration. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Explain why a self-financing trading strategy must be adapted before a pricing model can trust it. Teach-back prompt: explain Brownian Motion, Filtration, And Adapted Processes as a tool for asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations to someone who knows probability but not this course.

Chapter 2: Quadratic Variation And The Failure Of Ordinary Calculus

Why it matters: realized volatility, rough price paths, diffusion measurement, signal noise accumulation is hard to reason about without a model for random evolution. Brownian increments are small, but their squared increments add up to time rather than disappearing. Definition and statement: Along refining partitions, \(\sum_k(B_{t_{k+1}}-B_{t_k})^2\) converges in probability to t, while total variation is infinite. Procedure: Compare orders of magnitude: dB is size \(\sqrt{dt}\) , so \((dB)^2\) contributes like dt and \(dB\,dt\) or \((dt)^2\) disappear. Main formula or test: \((dB_t)^2=dt\) , \(dt\,dB_t=0\) , and \((dt)^2=0\) in Itô differential bookkeeping. Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \((dB_t)^2=dt\) , \(dt\,dB_t=0\) , and \((dt)^2=0\) in Itô differential bookkeeping.
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of realized volatility, rough price paths, diffusion measurement, signal noise accumulation. Classic trap: using ordinary chain rule and missing the quadratic variation correction Repair: write the Ito table before differentiating any function of Brownian motion Guided practice: Explain why \(d(B_t^2)\) is not just \(2B_t\,dB_t\) and identify the missing term. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Connect quadratic variation to realized volatility from high-frequency returns. Teach-back prompt: explain Quadratic Variation And The Failure Of Ordinary Calculus as a tool for realized volatility, rough price paths, diffusion measurement, signal noise accumulation to someone who knows probability but not this course.

Chapter 3: Ito Integral And Ito Isometry

Why it matters: noise-driven accumulated exposure, hedging error variance, stochastic signal filtering is hard to reason about without a model for random evolution. The Ito integral is a limit of sums that uses the information available before each noise increment arrives. Definition and statement: For square-integrable adapted H, \(\int_0^TH_t\,dB_t\) is the L2 limit of left-endpoint sums, with \(\mathbb{E}[(\int H\,dB)^2]=\mathbb{E}[\int H^2\,dt]\) . Procedure: Approximate by simple adapted steps, use independent Brownian increments for mean zero, apply Ito isometry for second moments, then pass to limits. Main formula or test: \(\mathbb{E}\!\left[\int_0^T H_t\,dB_t\right]=0\) and \(\mathbb{E}\!\left[\left(\int_0^T H_t\,dB_t\right)^2\right]=\mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]\) . Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \(\mathbb{E}\!\left[\int_0^T H_t\,dB_t\right]=0\) and \(\mathbb{E}\!\left[\left(\int_0^T H_t\,dB_t\right)^2\right]=\mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]\) .
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of noise-driven accumulated exposure, hedging error variance, stochastic signal filtering. Classic trap: treating dB like ordinary dt and pulling random future terms out of the integral Repair: check adaptedness and square integrability before applying the isometry Guided practice: Compute \(\mathbb{E}\!\left[\left(\int_0^TB_t\,dB_t\right)^2\right]\) using Ito isometry. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Compute the variance of accumulated noise exposure when \(H_t\) is a deterministic seasonal load. Teach-back prompt: explain Ito Integral And Ito Isometry as a tool for noise-driven accumulated exposure, hedging error variance, stochastic signal filtering to someone who knows probability but not this course.

Chapter 4: Ito Formula, Product Rule, And Exponential Martingales

Why it matters: log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance is hard to reason about without a model for random evolution. Ito formula is the chain rule with a volatility-curvature correction. Definition and statement: If \(dX_t=\mu_t\,dt+\sigma_t\,dB_t\) , then \(df(t,X_t)=(f_t+\mu f_x+\tfrac12\sigma^2f_{xx})\,dt+\sigma f_x\,dB_t\) . Procedure: Identify drift, diffusion, derivatives of f, insert into Ito formula, collect dt and dB terms, and interpret martingale versus drift parts. Main formula or test: \(d(fg)=f\,dg+g\,df+d[f,g]\) , and \(\exp(\theta B_t-\tfrac12\theta^2t)\) is a martingale. Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \(d(fg)=f\,dg+g\,df+d[f,g]\) , and \(\exp(\theta B_t-\tfrac12\theta^2t)\) is a martingale.
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance. Classic trap: forgetting that product rules include a quadratic covariation term Repair: compute the dX dY term before simplifying a product or chain rule Guided practice: Use Ito formula to find \(d(\log S_t)\) when \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) . Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Derive the dynamics of \(\exp(B_t-t/2)\) and explain why it has zero drift. Teach-back prompt: explain Ito Formula, Product Rule, And Exponential Martingales as a tool for log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance to someone who knows probability but not this course.

Chapter 5: SDEs, Drift, Diffusion, And Existence Intuition

Why it matters: population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials is hard to reason about without a model for random evolution. An SDE says the next tiny change is a predictable tendency plus a noise shock scaled by the current state. Definition and statement: An Ito SDE has form \(dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\) with adapted solution satisfying the integral equation. Procedure: State the integral form, identify drift and diffusion, check initial condition, note regularity assumptions, and explain the modeling meaning of each coefficient. Main formula or test: \(X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\) . Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \(X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\) .
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials. Classic trap: treating an SDE as an ordinary differential equation with a random term pasted on after solving Repair: write the integral equation and keep the stochastic integral inside the solution definition Guided practice: For \(dX_t=aX_t\,dt+bX_t\,dB_t\) , identify drift, diffusion, and why the solution is not \(e^{(a+b)t}\) . Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Choose drift and diffusion terms for a noisy mean-reverting temperature model. Teach-back prompt: explain SDEs, Drift, Diffusion, And Existence Intuition as a tool for population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials to someone who knows probability but not this course.

Chapter 6: Linear SDEs, Geometric Brownian Motion, And OU Models

Why it matters: stock prices, interest-rate factors, commodity spreads, physical velocity with friction is hard to reason about without a model for random evolution. Some SDEs have structure strong enough to solve explicitly after the right transform or integrating factor. Definition and statement: Geometric Brownian motion solves \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) with \(S_t=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\) . Procedure: Try log transform for multiplicative noise, integrating factor for linear SDEs, and mean/variance formulas for Gaussian mean-reverting models. Main formula or test: OU: \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) , with mean \(\theta+(X_0-\theta)e^{-\kappa t}\) . Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: OU: \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) , with mean \(\theta+(X_0-\theta)e^{-\kappa t}\) .
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of stock prices, interest-rate factors, commodity spreads, physical velocity with friction. Classic trap: forgetting the \(-\tfrac12\sigma^2\) correction in geometric Brownian motion log dynamics Repair: derive log \(S_t\) with Ito formula rather than guessing the exponential solution Guided practice: Solve geometric Brownian motion and compute E[\(S_t\) ]. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Use an OU model to explain mean-reverting short-rate behavior. Teach-back prompt: explain Linear SDEs, Geometric Brownian Motion, And OU Models as a tool for stock prices, interest-rate factors, commodity spreads, physical velocity with friction to someone who knows probability but not this course.

Chapter 7: Generators, Kolmogorov Equations, And Feynman-Kac

Why it matters: option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews is hard to reason about without a model for random evolution. The generator tells how expected future values start changing from the current state. Definition and statement: For \(dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t\) , generator \(\mathcal{L}f=bf_x+\tfrac12\sigma^2f_{xx}\) ; Feynman-Kac links parabolic PDEs to discounted expectations. Procedure: Compute L, choose the terminal or boundary payoff, write the backward equation, and interpret the solution as an expectation under the diffusion. Main formula or test: \(u_t+\mathcal{L}u-ru+g=0\) with terminal condition \(u(T,x)=\phi(x)\) has an expectation representation under standard assumptions. Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \(u_t+\mathcal{L}u-ru+g=0\) with terminal condition \(u(T,x)=\phi(x)\) has an expectation representation under standard assumptions.
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews. Classic trap: mixing forward density equations with backward value equations Repair: ask whether the unknown is a distribution moving forward or a value function looking backward from payoff Guided practice: Find the generator of geometric Brownian motion and write the backward pricing PDE for a terminal payoff. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Use a generator to approximate a diffusion hitting probability by solving a boundary value equation. Teach-back prompt: explain Generators, Kolmogorov Equations, And Feynman-Kac as a tool for option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews to someone who knows probability but not this course.

Chapter 8: Change Of Measure And Girsanov

Why it matters: risk-neutral pricing, likelihood ratios, filtering, rare-event simulation is hard to reason about without a model for random evolution. Girsanov changes the likelihood of paths so that what looked like drift under one measure can become part of Brownian motion under another. Definition and statement: Under suitable Novikov-type conditions, an exponential martingale defines a new measure under which \(B_t\) minus an integrated drift becomes Brownian motion. Procedure: Identify market price of risk or drift shift, build the exponential density, state the condition, transform Brownian motion, and rewrite the SDE under the new measure. Main formula or test: \(Z_t=\exp\!\left(-\int\theta\,dB-\tfrac12\int\theta^2\,dt\right)\) , and \(W_t=B_t+\int\theta\,dt\) is Brownian under the new measure. Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \(Z_t=\exp\!\left(-\int\theta\,dB-\tfrac12\int\theta^2\,dt\right)\) , and \(W_t=B_t+\int\theta\,dt\) is Brownian under the new measure.
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of risk-neutral pricing, likelihood ratios, filtering, rare-event simulation. Classic trap: changing drift without changing the probability measure or density process Repair: write the density process first and state which measure each Brownian motion belongs to Guided practice: Given \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) , choose \(\theta\) so discounted S has drift zero under the new measure. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Explain why risk-neutral pricing changes expected growth but not the volatility coefficient. Teach-back prompt: explain Change Of Measure And Girsanov as a tool for risk-neutral pricing, likelihood ratios, filtering, rare-event simulation to someone who knows probability but not this course.

Chapter 9: Black-Scholes Pricing And Hedging

Why it matters: equity options, hedging desks, implied volatility, risk management, model-risk analysis is hard to reason about without a model for random evolution. A continuously adjusted hedge can cancel Brownian risk and force the remaining position to earn the risk-free rate. Definition and statement: Under geometric Brownian stock dynamics with constant r and \(\sigma\) , option value V satisfies \(V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0\) with terminal payoff. Procedure: Apply Ito to V(t,\(S_t\) ), form delta hedge, cancel dB term, impose risk-free return, state terminal condition, and connect to risk-neutral expectation. Main formula or test: The call price is \(C=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)\) under the standard Black-Scholes assumptions. Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: The call price is \(C=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)\) under the standard Black-Scholes assumptions.
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of equity options, hedging desks, implied volatility, risk management, model-risk analysis. Classic trap: memorizing the formula without knowing the assumptions and hedge argument Repair: derive the PDE before using the closed-form call expression Guided practice: Derive the Black-Scholes PDE for a European option from Ito formula and a delta hedge. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Explain how higher volatility changes call option value through convex payoff exposure. Teach-back prompt: explain Black-Scholes Pricing And Hedging as a tool for equity options, hedging desks, implied volatility, risk management, model-risk analysis to someone who knows probability but not this course.

Chapter 10: Stopping, Barriers, And Continuous-Time Martingales

Why it matters: barrier options, ruin probabilities, hitting thresholds, sequential detection is hard to reason about without a model for random evolution. Stopping tools are powerful because martingales keep fair value, but only when the stopping conditions are legal. Definition and statement: For suitable stopping times tau and martingales \(M_t\) , optional sampling gives \(\mathbb{E}[M_\tau]=\mathbb{E}[M_0]\) under boundedness or integrability hypotheses. Procedure: Build a martingale from Brownian motion or an SDE, identify the stopping time, localize or bound it, apply optional sampling, and solve boundary equations. Main formula or test: \(B_t^2-t\) is a martingale; \(\exp(\theta B_t-\tfrac12\theta^2t)\) is also a martingale. Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \(B_t^2-t\) is a martingale; \(\exp(\theta B_t-\tfrac12\theta^2t)\) is also a martingale.
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of barrier options, ruin probabilities, hitting thresholds, sequential detection. Classic trap: using optional sampling at an infinite-mean hitting time without justification Repair: truncate the stopping time or state the theorem conditions before passing to a limit Guided practice: Use a Brownian martingale to compute the probability of hitting upper barrier a before lower barrier -b. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Price or bound a simple barrier-style payoff by identifying the relevant hitting event. Teach-back prompt: explain Stopping, Barriers, And Continuous-Time Martingales as a tool for barrier options, ruin probabilities, hitting thresholds, sequential detection to someone who knows probability but not this course.

Chapter 11: Euler-Maruyama Simulation And Numerical Error

Why it matters: risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation is hard to reason about without a model for random evolution. Simulation replaces infinitesimal drift and noise with small time steps; the model becomes useful only when the error story is honest. Definition and statement: Euler-Maruyama uses \(X_{n+1}=X_n+b(t_n,X_n)\Delta t+\sigma(t_n,X_n)\Delta B_n\) with \(\Delta B_n\sim\mathcal{N}(0,\Delta t)\) . Procedure: Choose time step, generate Brownian increments, update drift and diffusion, repeat many paths, estimate quantity, and separate Monte Carlo error from time-discretization error. Main formula or test: \(\Delta B_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0,1)\) . Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: \(\Delta B_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0,1)\) .
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation. Classic trap: using too-large time steps and mistaking discretization bias for model behavior Repair: halve the time step, increase path count, and compare stable quantities Guided practice: Write the Euler-Maruyama update for \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) and describe one convergence check. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Simulate a geometric Brownian portfolio loss distribution and report uncertainty in the estimate. Teach-back prompt: explain Euler-Maruyama Simulation And Numerical Error as a tool for risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation to someone who knows probability but not this course.

Chapter 12: Model Choice, Applications, And Readiness Synthesis

Why it matters: option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations is hard to reason about without a model for random evolution. Stochastic calculus is useful when the application has continuous-time noise, but the model only earns trust when assumptions and outputs are checked. Definition and statement: A stochastic-calculus model specifies filtration, driving noise, SDE coefficients, payoff or observable, measure, numerical method, and validation target. Procedure: Classify the problem, choose Brownian/SDE/tool family, state assumptions, compute or simulate, interpret units, run a trap check, and state model risk. Main formula or test: Model choice checklist: information, state, drift, diffusion, boundary, payoff, measure, numerical scheme, validation. Worked model:

1. First name the random object and the information available.
2. Match the problem to the formula: Model choice checklist: information, state, drift, diffusion, boundary, payoff, measure, numerical scheme, validation.
3. Carry out the calculation while keeping the assumption visible.
4. Interpret the answer in the setting of option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations. Classic trap: using Black-Scholes or GBM as a universal default instead of matching data and assumptions Repair: force a model-choice justification before any formula and list at least one failure mode Guided practice: Choose between GBM, OU, reflected diffusion, or a jump model for four short scenarios and justify one limitation of each choice. Independent practice: change the numbers, state the assumptions, solve, and interpret. Transfer: Design a small stochastic-calculus modeling memo for a hedging, risk, or noisy-physics application. Teach-back prompt: explain Model Choice, Applications, And Readiness Synthesis as a tool for option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations to someone who knows probability but not this course.

Problem Set

Problem Set Questions

Problems are original mixed-practice tasks for the course scope.

Chapter 1: Brownian Motion, Filtration, And Adapted Processes

1. Recall and setup: state the definition behind Brownian Motion, Filtration, And Adapted Processes and identify all assumptions needed before using \(B_t-B_s\) is independent of \(\mathcal{F}_s\) , and \(B_t-B_s\sim\mathcal{N}(0,t-s)\) for \(s<t\) under the Brownian filtration.
2. Guided computation: Decide whether \(H_t=B_{t/2}\) , \(H_t=B_{t+1}\) , and \(H_t=\mathbf{1}_{\{B_t>0\}}\) are adapted to the Brownian filtration.
3. Application transfer: build a model for asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations, solve one representative quantity, and explain the result in units or probabilities.
4. Trap repair: a student tries letting a trading strategy or integrand depend on tomorrow Brownian increment. Diagnose the mistake and rewrite the first two correct steps.
5. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 2: Quadratic Variation And The Failure Of Ordinary Calculus

6. Recall and setup: state the definition behind Quadratic Variation And The Failure Of Ordinary Calculus and identify all assumptions needed before using \((dB_t)^2=dt\) , \(dt\,dB_t=0\) , and \((dt)^2=0\) in Itô differential bookkeeping.
7. Guided computation: Explain why \(d(B_t^2)\) is not just \(2B_t\,dB_t\) and identify the missing term.
8. Application transfer: build a model for realized volatility, rough price paths, diffusion measurement, signal noise accumulation, solve one representative quantity, and explain the result in units or probabilities.
9. Trap repair: a student tries using ordinary chain rule and missing the quadratic variation correction. Diagnose the mistake and rewrite the first two correct steps.
10. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 3: Ito Integral And Ito Isometry

11. Recall and setup: state the definition behind Ito Integral And Ito Isometry and identify all assumptions needed before using \(\mathbb{E}\!\left[\int_0^T H_t\,dB_t\right]=0\) and \(\mathbb{E}\!\left[\left(\int_0^T H_t\,dB_t\right)^2\right]=\mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]\) .
12. Guided computation: Compute \(\mathbb{E}\!\left[\left(\int_0^TB_t\,dB_t\right)^2\right]\) using Ito isometry.
13. Application transfer: build a model for noise-driven accumulated exposure, hedging error variance, stochastic signal filtering, solve one representative quantity, and explain the result in units or probabilities.
14. Trap repair: a student tries treating dB like ordinary dt and pulling random future terms out of the integral. Diagnose the mistake and rewrite the first two correct steps.
15. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 4: Ito Formula, Product Rule, And Exponential Martingales

16. Recall and setup: state the definition behind Ito Formula, Product Rule, And Exponential Martingales and identify all assumptions needed before using \(d(fg)=f\,dg+g\,df+d[f,g]\) , and \(\exp(\theta B_t-\tfrac12\theta^2t)\) is a martingale.
17. Guided computation: Use Ito formula to find \(d(\log S_t)\) when \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) .
18. Application transfer: build a model for log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance, solve one representative quantity, and explain the result in units or probabilities.
19. Trap repair: a student tries forgetting that product rules include a quadratic covariation term. Diagnose the mistake and rewrite the first two correct steps.
20. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 5: SDEs, Drift, Diffusion, And Existence Intuition

21. Recall and setup: state the definition behind SDEs, Drift, Diffusion, And Existence Intuition and identify all assumptions needed before using \(X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\) .
22. Guided computation: For \(dX_t=aX_t\,dt+bX_t\,dB_t\) , identify drift, diffusion, and why the solution is not \(e^{(a+b)t}\) .
23. Application transfer: build a model for population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials, solve one representative quantity, and explain the result in units or probabilities.
24. Trap repair: a student tries treating an SDE as an ordinary differential equation with a random term pasted on after solving. Diagnose the mistake and rewrite the first two correct steps.
25. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 6: Linear SDEs, Geometric Brownian Motion, And OU Models

26. Recall and setup: state the definition behind Linear SDEs, Geometric Brownian Motion, And OU Models and identify all assumptions needed before using OU: \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) , with mean \(\theta+(X_0-\theta)e^{-\kappa t}\) .
27. Guided computation: Solve geometric Brownian motion and compute E[\(S_t\) ].
28. Application transfer: build a model for stock prices, interest-rate factors, commodity spreads, physical velocity with friction, solve one representative quantity, and explain the result in units or probabilities.
29. Trap repair: a student tries forgetting the \(-\tfrac12\sigma^2\) correction in geometric Brownian motion log dynamics. Diagnose the mistake and rewrite the first two correct steps.
30. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 7: Generators, Kolmogorov Equations, And Feynman-Kac

31. Recall and setup: state the definition behind Generators, Kolmogorov Equations, And Feynman-Kac and identify all assumptions needed before using \(u_t+\mathcal{L}u-ru+g=0\) with terminal condition \(u(T,x)=\phi(x)\) has an expectation representation under standard assumptions.
32. Guided computation: Find the generator of geometric Brownian motion and write the backward pricing PDE for a terminal payoff.
33. Application transfer: build a model for option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews, solve one representative quantity, and explain the result in units or probabilities.
34. Trap repair: a student tries mixing forward density equations with backward value equations. Diagnose the mistake and rewrite the first two correct steps.
35. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 8: Change Of Measure And Girsanov

36. Recall and setup: state the definition behind Change Of Measure And Girsanov and identify all assumptions needed before using \(Z_t=\exp\!\left(-\int\theta\,dB-\tfrac12\int\theta^2\,dt\right)\) , and \(W_t=B_t+\int\theta\,dt\) is Brownian under the new measure.
37. Guided computation: Given \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) , choose \(\theta\) so discounted S has drift zero under the new measure.
38. Application transfer: build a model for risk-neutral pricing, likelihood ratios, filtering, rare-event simulation, solve one representative quantity, and explain the result in units or probabilities.
39. Trap repair: a student tries changing drift without changing the probability measure or density process. Diagnose the mistake and rewrite the first two correct steps.
40. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 9: Black-Scholes Pricing And Hedging

41. Recall and setup: state the definition behind Black-Scholes Pricing And Hedging and identify all assumptions needed before using The call price is \(C=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)\) under the standard Black-Scholes assumptions.
42. Guided computation: Derive the Black-Scholes PDE for a European option from Ito formula and a delta hedge.
43. Application transfer: build a model for equity options, hedging desks, implied volatility, risk management, model-risk analysis, solve one representative quantity, and explain the result in units or probabilities.
44. Trap repair: a student tries memorizing the formula without knowing the assumptions and hedge argument. Diagnose the mistake and rewrite the first two correct steps.
45. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 10: Stopping, Barriers, And Continuous-Time Martingales

46. Recall and setup: state the definition behind Stopping, Barriers, And Continuous-Time Martingales and identify all assumptions needed before using \(B_t^2-t\) is a martingale; \(\exp(\theta B_t-\tfrac12\theta^2t)\) is also a martingale.
47. Guided computation: Use a Brownian martingale to compute the probability of hitting upper barrier a before lower barrier -b.
48. Application transfer: build a model for barrier options, ruin probabilities, hitting thresholds, sequential detection, solve one representative quantity, and explain the result in units or probabilities.
49. Trap repair: a student tries using optional sampling at an infinite-mean hitting time without justification. Diagnose the mistake and rewrite the first two correct steps.
50. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 11: Euler-Maruyama Simulation And Numerical Error

51. Recall and setup: state the definition behind Euler-Maruyama Simulation And Numerical Error and identify all assumptions needed before using \(\Delta B_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0,1)\) .
52. Guided computation: Write the Euler-Maruyama update for \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) and describe one convergence check.
53. Application transfer: build a model for risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation, solve one representative quantity, and explain the result in units or probabilities.
54. Trap repair: a student tries using too-large time steps and mistaking discretization bias for model behavior. Diagnose the mistake and rewrite the first two correct steps.
55. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Chapter 12: Model Choice, Applications, And Readiness Synthesis

56. Recall and setup: state the definition behind Model Choice, Applications, And Readiness Synthesis and identify all assumptions needed before using Model choice checklist: information, state, drift, diffusion, boundary, payoff, measure, numerical scheme, validation.
57. Guided computation: Choose between GBM, OU, reflected diffusion, or a jump model for four short scenarios and justify one limitation of each choice.
58. Application transfer: build a model for option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations, solve one representative quantity, and explain the result in units or probabilities.
59. Trap repair: a student tries using Black-Scholes or GBM as a universal default instead of matching data and assumptions. Diagnose the mistake and rewrite the first two correct steps.
60. Teach-back: explain why this milestone’s method is preferred over a neighboring stochastic model for this scenario.

Cumulative Application Problems

A1. Model-choice application: for Black-Scholes option pricing, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A2. Model-choice application: for delta hedging, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A3. Model-choice application: for risk-neutral valuation, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A4. Model-choice application: for interest-rate mean reversion, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A5. Model-choice application: for realized volatility, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A6. Model-choice application: for barrier risk, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A7. Model-choice application: for Monte Carlo risk simulation, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A8. Model-choice application: for diffusion physics, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A9. Model-choice application: for stochastic climate perturbations, specify the state, time index, randomness source, assumptions, calculation target, and one limitation. A10. Model-choice application: for noisy population models, specify the state, time index, randomness source, assumptions, calculation target, and one limitation.

Worked Solutions

Worked Solutions

Solutions show setup, calculation, interpretation, and repair. They are intentionally fuller than answer keys.

Chapter 1: Brownian Motion, Filtration, And Adapted Processes

Solution 1: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: Brownian motion \(B_t\) is adapted to a filtration \(\mathcal{F}_t\) , and a process \(H_t\) is adapted when \(H_t\) is \(\mathcal{F}_t\) -measurable for each t. Solution 2: \(B_{t/2}\) and \(\mathbf{1}_{\{B_t>0\}}\) are adapted; \(B_{t+1}\) is not because it uses future information. The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 3: In asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations, choose the state and time index first. Then use \(B_t-B_s\) is independent of \(\mathcal{F}_s\) , and \(B_t-B_s\sim\mathcal{N}(0,t-s)\) for \(s<t\) under the Brownian filtration. only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 4: The mistake is letting a trading strategy or integrand depend on tomorrow Brownian increment. Repair by: ask what is known at time t and reject any rule that uses future noise A corrected solution restates the model and then recomputes the requested quantity. Solution 5: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 2: Quadratic Variation And The Failure Of Ordinary Calculus

Solution 6: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: Along refining partitions, \(\sum_k(B_{t_{k+1}}-B_{t_k})^2\) converges in probability to t, while total variation is infinite. Solution 7: Ito formula gives \(d(B_t^2)=2B_t\,dB_t+dt\) because Brownian quadratic variation contributes dt. The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 8: In realized volatility, rough price paths, diffusion measurement, signal noise accumulation, choose the state and time index first. Then use \((dB_t)^2=dt\) , \(dt\,dB_t=0\) , and \((dt)^2=0\) in Itô differential bookkeeping. only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 9: The mistake is using ordinary chain rule and missing the quadratic variation correction. Repair by: write the Ito table before differentiating any function of Brownian motion A corrected solution restates the model and then recomputes the requested quantity. Solution 10: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 3: Ito Integral And Ito Isometry

Solution 11: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: For square-integrable adapted H, \(\int_0^TH_t\,dB_t\) is the L2 limit of left-endpoint sums, with \(\mathbb{E}[(\int H\,dB)^2]=\mathbb{E}[\int H^2\,dt]\) . Solution 12: Use \(\mathbb{E}\!\left[\int_0^TB_t^2\,dt\right]=\int_0^T\mathbb{E}[B_t^2]\,dt=\int_0^Tt\,dt=\dfrac{T^2}{2}\) . The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 13: In noise-driven accumulated exposure, hedging error variance, stochastic signal filtering, choose the state and time index first. Then use \(\mathbb{E}\!\left[\int_0^T H_t\,dB_t\right]=0\) and \(\mathbb{E}\!\left[\left(\int_0^T H_t\,dB_t\right)^2\right]=\mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]\) . only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 14: The mistake is treating dB like ordinary dt and pulling random future terms out of the integral. Repair by: check adaptedness and square integrability before applying the isometry A corrected solution restates the model and then recomputes the requested quantity. Solution 15: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 4: Ito Formula, Product Rule, And Exponential Martingales

Solution 16: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: If \(dX_t=\mu_t\,dt+\sigma_t\,dB_t\) , then \(df(t,X_t)=(f_t+\mu f_x+\tfrac12\sigma^2f_{xx})\,dt+\sigma f_x\,dB_t\) . Solution 17: \(d\log S_t=(\mu-\tfrac12\sigma^2)\,dt+\sigma\,dB_t\) . The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 18: In log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance, choose the state and time index first. Then use \(d(fg)=f\,dg+g\,df+d[f,g]\) , and \(\exp(\theta B_t-\tfrac12\theta^2t)\) is a martingale. only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 19: The mistake is forgetting that product rules include a quadratic covariation term. Repair by: compute the dX dY term before simplifying a product or chain rule A corrected solution restates the model and then recomputes the requested quantity. Solution 20: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 5: SDEs, Drift, Diffusion, And Existence Intuition

Solution 21: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: An Ito SDE has form \(dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\) with adapted solution satisfying the integral equation. Solution 22: Drift is aX_t, diffusion is bX_t, and Ito correction appears in the exponential solution. The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 23: In population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials, choose the state and time index first. Then use \(X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\) . only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 24: The mistake is treating an SDE as an ordinary differential equation with a random term pasted on after solving. Repair by: write the integral equation and keep the stochastic integral inside the solution definition A corrected solution restates the model and then recomputes the requested quantity. Solution 25: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 6: Linear SDEs, Geometric Brownian Motion, And OU Models

Solution 26: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: Geometric Brownian motion solves \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) with \(S_t=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\) . Solution 27: \(S_t=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\) and \(\mathbb{E}[S_t]=S_0e^{\mu t}\) . The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 28: In stock prices, interest-rate factors, commodity spreads, physical velocity with friction, choose the state and time index first. Then use OU: \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) , with mean \(\theta+(X_0-\theta)e^{-\kappa t}\) . only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 29: The mistake is forgetting the \(-\tfrac12\sigma^2\) correction in geometric Brownian motion log dynamics. Repair by: derive log \(S_t\) with Ito formula rather than guessing the exponential solution A corrected solution restates the model and then recomputes the requested quantity. Solution 30: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 7: Generators, Kolmogorov Equations, And Feynman-Kac

Solution 31: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: For \(dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t\) , generator \(\mathcal{L}f=bf_x+\tfrac12\sigma^2f_{xx}\) ; Feynman-Kac links parabolic PDEs to discounted expectations. Solution 32: \(\mathcal{L}f=\mu xf_x+\tfrac12\sigma^2x^2f_{xx}\) ; the value solves \(u_t+\mathcal{L}u-ru=0\) with terminal payoff. The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 33: In option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews, choose the state and time index first. Then use \(u_t+\mathcal{L}u-ru+g=0\) with terminal condition \(u(T,x)=\phi(x)\) has an expectation representation under standard assumptions. only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 34: The mistake is mixing forward density equations with backward value equations. Repair by: ask whether the unknown is a distribution moving forward or a value function looking backward from payoff A corrected solution restates the model and then recomputes the requested quantity. Solution 35: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 8: Change Of Measure And Girsanov

Solution 36: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: Under suitable Novikov-type conditions, an exponential martingale defines a new measure under which \(B_t\) minus an integrated drift becomes Brownian motion. Solution 37: Use \(\theta=(\mu-r)/\sigma\) with the correct sign convention and rewrite \(dS_t=rS_t\,dt+\sigma S_t\,dW_t\) under Q. The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 38: In risk-neutral pricing, likelihood ratios, filtering, rare-event simulation, choose the state and time index first. Then use \(Z_t=\exp\!\left(-\int\theta\,dB-\tfrac12\int\theta^2\,dt\right)\) , and \(W_t=B_t+\int\theta\,dt\) is Brownian under the new measure. only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 39: The mistake is changing drift without changing the probability measure or density process. Repair by: write the density process first and state which measure each Brownian motion belongs to A corrected solution restates the model and then recomputes the requested quantity. Solution 40: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 9: Black-Scholes Pricing And Hedging

Solution 41: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: Under geometric Brownian stock dynamics with constant r and \(\sigma\) , option value V satisfies \(V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0\) with terminal payoff. Solution 42: Compute dV, hold \(\Delta=V_S\) shares to cancel dB, and require the riskless residual to grow at r. The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 43: In equity options, hedging desks, implied volatility, risk management, model-risk analysis, choose the state and time index first. Then use The call price is \(C=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)\) under the standard Black-Scholes assumptions. only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 44: The mistake is memorizing the formula without knowing the assumptions and hedge argument. Repair by: derive the PDE before using the closed-form call expression A corrected solution restates the model and then recomputes the requested quantity. Solution 45: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 10: Stopping, Barriers, And Continuous-Time Martingales

Solution 46: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: For suitable stopping times tau and martingales \(M_t\) , optional sampling gives \(\mathbb{E}[M_\tau]=\mathbb{E}[M_0]\) under boundedness or integrability hypotheses. Solution 47: Stop \(B_t\) at the bounded hitting time of the interval endpoints and solve \(0=ap-b(1-p)\) . The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 48: In barrier options, ruin probabilities, hitting thresholds, sequential detection, choose the state and time index first. Then use \(B_t^2-t\) is a martingale; \(\exp(\theta B_t-\tfrac12\theta^2t)\) is also a martingale. only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 49: The mistake is using optional sampling at an infinite-mean hitting time without justification. Repair by: truncate the stopping time or state the theorem conditions before passing to a limit A corrected solution restates the model and then recomputes the requested quantity. Solution 50: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 11: Euler-Maruyama Simulation And Numerical Error

Solution 51: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: Euler-Maruyama uses \(X_{n+1}=X_n+b(t_n,X_n)\Delta t+\sigma(t_n,X_n)\Delta B_n\) with \(\Delta B_n\sim\mathcal{N}(0,\Delta t)\) . Solution 52: \(X_{n+1}=X_n+\kappa(\theta-X_n)\Delta t+\sigma\sqrt{\Delta t}Z_n\) ; compare estimates across smaller Delta t. The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 53: In risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation, choose the state and time index first. Then use \(\Delta B_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0,1)\) . only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 54: The mistake is using too-large time steps and mistaking discretization bias for model behavior. Repair by: halve the time step, increase path count, and compare stable quantities A corrected solution restates the model and then recomputes the requested quantity. Solution 55: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Chapter 12: Model Choice, Applications, And Readiness Synthesis

Solution 56: A correct setup names the stochastic object, the information available, and the assumptions. The relevant statement is: A stochastic-calculus model specifies filtration, driving noise, SDE coefficients, payoff or observable, measure, numerical method, and validation target. Solution 57: Match multiplicative growth to GBM, mean reversion to OU, boundaries to reflected or killed diffusions, and discontinuous shocks to jump models outside this course boundary. The important exam-room habit is to show the intermediate stochastic meaning, not only the final expression. Solution 58: In option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations, choose the state and time index first. Then use Model choice checklist: information, state, drift, diffusion, boundary, payoff, measure, numerical scheme, validation. only after the assumptions match. Interpret the result as a prediction, probability, rate, expectation, or model diagnostic as appropriate. Solution 59: The mistake is using Black-Scholes or GBM as a universal default instead of matching data and assumptions. Repair by: force a model-choice justification before any formula and list at least one failure mode A corrected solution restates the model and then recomputes the requested quantity. Solution 60: The teach-back should include the model, assumptions, calculation path, result, and limitation. A strong answer also names one neighboring model that would be wrong and why.

Retrieval Review Pack

Retrieval And Mixed Review Pack

Use this pack after delays. Attempt recall before rereading.

Review 1: Brownian Motion, Filtration, And Adapted Processes

• First-exposure recall: state the main idea without notes: Stochastic calculus starts by saying what noise is visible now and forbidding the integrand from seeing the future.
• Immediate practice: Decide whether \(H_t=B_{t/2}\) , \(H_t=B_{t+1}\) , and \(H_t=\mathbf{1}_{\{B_t>0\}}\) are adapted to the Brownian filtration.
• Near-transfer: Explain why a self-financing trading strategy must be adapted before a pricing model can trust it.
• Delayed-review trap: identify and repair letting a trading strategy or integrand depend on tomorrow Brownian increment.
• Mixed-review selector: decide whether a new problem calls for Brownian Motion, Filtration, And Adapted Processes or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use asset price shocks, sensor-noise models, diffusion physics, interest-rate perturbations to explain the idea.

Review 2: Quadratic Variation And The Failure Of Ordinary Calculus

• First-exposure recall: state the main idea without notes: Brownian increments are small, but their squared increments add up to time rather than disappearing.
• Immediate practice: Explain why \(d(B_t^2)\) is not just \(2B_t\,dB_t\) and identify the missing term.
• Near-transfer: Connect quadratic variation to realized volatility from high-frequency returns.
• Delayed-review trap: identify and repair using ordinary chain rule and missing the quadratic variation correction.
• Mixed-review selector: decide whether a new problem calls for Quadratic Variation And The Failure Of Ordinary Calculus or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use realized volatility, rough price paths, diffusion measurement, signal noise accumulation to explain the idea.

Review 3: Ito Integral And Ito Isometry

• First-exposure recall: state the main idea without notes: The Ito integral is a limit of sums that uses the information available before each noise increment arrives.
• Immediate practice: Compute \(\mathbb{E}\!\left[\left(\int_0^TB_t\,dB_t\right)^2\right]\) using Ito isometry.
• Near-transfer: Compute the variance of accumulated noise exposure when \(H_t\) is a deterministic seasonal load.
• Delayed-review trap: identify and repair treating dB like ordinary dt and pulling random future terms out of the integral.
• Mixed-review selector: decide whether a new problem calls for Ito Integral And Ito Isometry or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use noise-driven accumulated exposure, hedging error variance, stochastic signal filtering to explain the idea.

Review 4: Ito Formula, Product Rule, And Exponential Martingales

• First-exposure recall: state the main idea without notes: Ito formula is the chain rule with a volatility-curvature correction.
• Immediate practice: Use Ito formula to find \(d(\log S_t)\) when \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) .
• Near-transfer: Derive the dynamics of \(\exp(B_t-t/2)\) and explain why it has zero drift.
• Delayed-review trap: identify and repair forgetting that product rules include a quadratic covariation term.
• Mixed-review selector: decide whether a new problem calls for Ito Formula, Product Rule, And Exponential Martingales or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use log price dynamics, option payoff transformations, likelihood ratios, stochastic energy balance to explain the idea.

Review 5: SDEs, Drift, Diffusion, And Existence Intuition

• First-exposure recall: state the main idea without notes: An SDE says the next tiny change is a predictable tendency plus a noise shock scaled by the current state.
• Immediate practice: For \(dX_t=aX_t\,dt+bX_t\,dB_t\) , identify drift, diffusion, and why the solution is not \(e^{(a+b)t}\) .
• Near-transfer: Choose drift and diffusion terms for a noisy mean-reverting temperature model.
• Delayed-review trap: identify and repair treating an SDE as an ordinary differential equation with a random term pasted on after solving.
• Mixed-review selector: decide whether a new problem calls for SDEs, Drift, Diffusion, And Existence Intuition or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use population with random growth shocks, temperature with noisy forcing, asset prices, neuron membrane potentials to explain the idea.

Review 6: Linear SDEs, Geometric Brownian Motion, And OU Models

• First-exposure recall: state the main idea without notes: Some SDEs have structure strong enough to solve explicitly after the right transform or integrating factor.
• Immediate practice: Solve geometric Brownian motion and compute E[\(S_t\) ].
• Near-transfer: Use an OU model to explain mean-reverting short-rate behavior.
• Delayed-review trap: identify and repair forgetting the \(-\tfrac12\sigma^2\) correction in geometric Brownian motion log dynamics.
• Mixed-review selector: decide whether a new problem calls for Linear SDEs, Geometric Brownian Motion, And OU Models or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use stock prices, interest-rate factors, commodity spreads, physical velocity with friction to explain the idea.

Review 7: Generators, Kolmogorov Equations, And Feynman-Kac

• First-exposure recall: state the main idea without notes: The generator tells how expected future values start changing from the current state.
• Immediate practice: Find the generator of geometric Brownian motion and write the backward pricing PDE for a terminal payoff.
• Near-transfer: Use a generator to approximate a diffusion hitting probability by solving a boundary value equation.
• Delayed-review trap: identify and repair mixing forward density equations with backward value equations.
• Mixed-review selector: decide whether a new problem calls for Generators, Kolmogorov Equations, And Feynman-Kac or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use option pricing PDEs, heat equation, diffusion hitting probabilities, optimal stopping previews to explain the idea.

Review 8: Change Of Measure And Girsanov

• First-exposure recall: state the main idea without notes: Girsanov changes the likelihood of paths so that what looked like drift under one measure can become part of Brownian motion under another.
• Immediate practice: Given \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) , choose \(\theta\) so discounted S has drift zero under the new measure.
• Near-transfer: Explain why risk-neutral pricing changes expected growth but not the volatility coefficient.
• Delayed-review trap: identify and repair changing drift without changing the probability measure or density process.
• Mixed-review selector: decide whether a new problem calls for Change Of Measure And Girsanov or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use risk-neutral pricing, likelihood ratios, filtering, rare-event simulation to explain the idea.

Review 9: Black-Scholes Pricing And Hedging

• First-exposure recall: state the main idea without notes: A continuously adjusted hedge can cancel Brownian risk and force the remaining position to earn the risk-free rate.
• Immediate practice: Derive the Black-Scholes PDE for a European option from Ito formula and a delta hedge.
• Near-transfer: Explain how higher volatility changes call option value through convex payoff exposure.
• Delayed-review trap: identify and repair memorizing the formula without knowing the assumptions and hedge argument.
• Mixed-review selector: decide whether a new problem calls for Black-Scholes Pricing And Hedging or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use equity options, hedging desks, implied volatility, risk management, model-risk analysis to explain the idea.

Review 10: Stopping, Barriers, And Continuous-Time Martingales

• First-exposure recall: state the main idea without notes: Stopping tools are powerful because martingales keep fair value, but only when the stopping conditions are legal.
• Immediate practice: Use a Brownian martingale to compute the probability of hitting upper barrier a before lower barrier -b.
• Near-transfer: Price or bound a simple barrier-style payoff by identifying the relevant hitting event.
• Delayed-review trap: identify and repair using optional sampling at an infinite-mean hitting time without justification.
• Mixed-review selector: decide whether a new problem calls for Stopping, Barriers, And Continuous-Time Martingales or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use barrier options, ruin probabilities, hitting thresholds, sequential detection to explain the idea.

Review 11: Euler-Maruyama Simulation And Numerical Error

• First-exposure recall: state the main idea without notes: Simulation replaces infinitesimal drift and noise with small time steps; the model becomes useful only when the error story is honest.
• Immediate practice: Write the Euler-Maruyama update for \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) and describe one convergence check.
• Near-transfer: Simulate a geometric Brownian portfolio loss distribution and report uncertainty in the estimate.
• Delayed-review trap: identify and repair using too-large time steps and mistaking discretization bias for model behavior.
• Mixed-review selector: decide whether a new problem calls for Euler-Maruyama Simulation And Numerical Error or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use risk simulation, scenario generation, stochastic control prototypes, physical diffusion computation to explain the idea.

Review 12: Model Choice, Applications, And Readiness Synthesis

• First-exposure recall: state the main idea without notes: Stochastic calculus is useful when the application has continuous-time noise, but the model only earns trust when assumptions and outputs are checked.
• Immediate practice: Choose between GBM, OU, reflected diffusion, or a jump model for four short scenarios and justify one limitation of each choice.
• Near-transfer: Design a small stochastic-calculus modeling memo for a hedging, risk, or noisy-physics application.
• Delayed-review trap: identify and repair using Black-Scholes or GBM as a universal default instead of matching data and assumptions.
• Mixed-review selector: decide whether a new problem calls for Model Choice, Applications, And Readiness Synthesis or a neighboring course model, and justify the choice.
• Final-synthesis teach-back: use option pricing, rates, credit intensity, climate diffusion, neuronal noise, queue diffusion approximations to explain the idea.

Proof And Counterexample Pack

Derivation, Proof, And Counterexample Pack

This pack keeps theorem conditions, proof ideas, derivations, and counterexamples visible for exam-prepared study.

Derivation 1: Brownian Motion, Filtration, And Adapted Processes

Statement to justify: Brownian motion \(B_t\) is adapted to a filtration \(\mathcal{F}_t\) , and a process \(H_t\) is adapted when \(H_t\) is \(\mathcal{F}_t\) -measurable for each t. Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \(B_t-B_s\) is independent of \(\mathcal{F}_s\) , and \(B_t-B_s\sim\mathcal{N}(0,t-s)\) for \(s<t\) under the Brownian filtration. Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries letting a trading strategy or integrand depend on tomorrow Brownian increment, construct a small example where the conclusion fails. Repair move: ask what is known at time t and reject any rule that uses future noise Exam variant: Decide whether \(H_t=B_{t/2}\) , \(H_t=B_{t+1}\) , and \(H_t=\mathbf{1}_{\{B_t>0\}}\) are adapted to the Brownian filtration.

Derivation 2: Quadratic Variation And The Failure Of Ordinary Calculus

Statement to justify: Along refining partitions, \(\sum_k(B_{t_{k+1}}-B_{t_k})^2\) converges in probability to t, while total variation is infinite. Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \((dB_t)^2=dt\) , \(dt\,dB_t=0\) , and \((dt)^2=0\) in Itô differential bookkeeping. Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries using ordinary chain rule and missing the quadratic variation correction, construct a small example where the conclusion fails. Repair move: write the Ito table before differentiating any function of Brownian motion Exam variant: Explain why \(d(B_t^2)\) is not just \(2B_t\,dB_t\) and identify the missing term.

Derivation 3: Ito Integral And Ito Isometry

Statement to justify: For square-integrable adapted H, \(\int_0^TH_t\,dB_t\) is the L2 limit of left-endpoint sums, with \(\mathbb{E}[(\int H\,dB)^2]=\mathbb{E}[\int H^2\,dt]\) . Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \(\mathbb{E}\!\left[\int_0^T H_t\,dB_t\right]=0\) and \(\mathbb{E}\!\left[\left(\int_0^T H_t\,dB_t\right)^2\right]=\mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]\) . Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries treating dB like ordinary dt and pulling random future terms out of the integral, construct a small example where the conclusion fails. Repair move: check adaptedness and square integrability before applying the isometry Exam variant: Compute \(\mathbb{E}\!\left[\left(\int_0^TB_t\,dB_t\right)^2\right]\) using Ito isometry.

Derivation 4: Ito Formula, Product Rule, And Exponential Martingales

Statement to justify: If \(dX_t=\mu_t\,dt+\sigma_t\,dB_t\) , then \(df(t,X_t)=(f_t+\mu f_x+\tfrac12\sigma^2f_{xx})\,dt+\sigma f_x\,dB_t\) . Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \(d(fg)=f\,dg+g\,df+d[f,g]\) , and \(\exp(\theta B_t-\tfrac12\theta^2t)\) is a martingale. Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries forgetting that product rules include a quadratic covariation term, construct a small example where the conclusion fails. Repair move: compute the dX dY term before simplifying a product or chain rule Exam variant: Use Ito formula to find \(d(\log S_t)\) when \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) .

Derivation 5: SDEs, Drift, Diffusion, And Existence Intuition

Statement to justify: An Ito SDE has form \(dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\) with adapted solution satisfying the integral equation. Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \(X_t=X_0+\int_0^t b(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\) . Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries treating an SDE as an ordinary differential equation with a random term pasted on after solving, construct a small example where the conclusion fails. Repair move: write the integral equation and keep the stochastic integral inside the solution definition Exam variant: For \(dX_t=aX_t\,dt+bX_t\,dB_t\) , identify drift, diffusion, and why the solution is not \(e^{(a+b)t}\) .

Derivation 6: Linear SDEs, Geometric Brownian Motion, And OU Models

Statement to justify: Geometric Brownian motion solves \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) with \(S_t=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\) . Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to OU: \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) , with mean \(\theta+(X_0-\theta)e^{-\kappa t}\) . Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries forgetting the \(-\tfrac12\sigma^2\) correction in geometric Brownian motion log dynamics, construct a small example where the conclusion fails. Repair move: derive log \(S_t\) with Ito formula rather than guessing the exponential solution Exam variant: Solve geometric Brownian motion and compute E[\(S_t\) ].

Derivation 7: Generators, Kolmogorov Equations, And Feynman-Kac

Statement to justify: For \(dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t\) , generator \(\mathcal{L}f=bf_x+\tfrac12\sigma^2f_{xx}\) ; Feynman-Kac links parabolic PDEs to discounted expectations. Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \(u_t+\mathcal{L}u-ru+g=0\) with terminal condition \(u(T,x)=\phi(x)\) has an expectation representation under standard assumptions. Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries mixing forward density equations with backward value equations, construct a small example where the conclusion fails. Repair move: ask whether the unknown is a distribution moving forward or a value function looking backward from payoff Exam variant: Find the generator of geometric Brownian motion and write the backward pricing PDE for a terminal payoff.

Derivation 8: Change Of Measure And Girsanov

Statement to justify: Under suitable Novikov-type conditions, an exponential martingale defines a new measure under which \(B_t\) minus an integrated drift becomes Brownian motion. Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \(Z_t=\exp\!\left(-\int\theta\,dB-\tfrac12\int\theta^2\,dt\right)\) , and \(W_t=B_t+\int\theta\,dt\) is Brownian under the new measure. Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries changing drift without changing the probability measure or density process, construct a small example where the conclusion fails. Repair move: write the density process first and state which measure each Brownian motion belongs to Exam variant: Given \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) , choose \(\theta\) so discounted S has drift zero under the new measure.

Derivation 9: Black-Scholes Pricing And Hedging

Statement to justify: Under geometric Brownian stock dynamics with constant r and \(\sigma\) , option value V satisfies \(V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0\) with terminal payoff. Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to The call price is \(C=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)\) under the standard Black-Scholes assumptions. Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries memorizing the formula without knowing the assumptions and hedge argument, construct a small example where the conclusion fails. Repair move: derive the PDE before using the closed-form call expression Exam variant: Derive the Black-Scholes PDE for a European option from Ito formula and a delta hedge.

Derivation 10: Stopping, Barriers, And Continuous-Time Martingales

Statement to justify: For suitable stopping times tau and martingales \(M_t\) , optional sampling gives \(\mathbb{E}[M_\tau]=\mathbb{E}[M_0]\) under boundedness or integrability hypotheses. Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \(B_t^2-t\) is a martingale; \(\exp(\theta B_t-\tfrac12\theta^2t)\) is also a martingale. Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries using optional sampling at an infinite-mean hitting time without justification, construct a small example where the conclusion fails. Repair move: truncate the stopping time or state the theorem conditions before passing to a limit Exam variant: Use a Brownian martingale to compute the probability of hitting upper barrier a before lower barrier -b.

Derivation 11: Euler-Maruyama Simulation And Numerical Error

Statement to justify: Euler-Maruyama uses \(X_{n+1}=X_n+b(t_n,X_n)\Delta t+\sigma(t_n,X_n)\Delta B_n\) with \(\Delta B_n\sim\mathcal{N}(0,\Delta t)\) . Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to \(\Delta B_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0,1)\) . Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries using too-large time steps and mistaking discretization bias for model behavior, construct a small example where the conclusion fails. Repair move: halve the time step, increase path count, and compare stable quantities Exam variant: Write the Euler-Maruyama update for \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dB_t\) and describe one convergence check.

Derivation 12: Model Choice, Applications, And Readiness Synthesis

Statement to justify: A stochastic-calculus model specifies filtration, driving noise, SDE coefficients, payoff or observable, measure, numerical method, and validation target. Proof or derivation idea: start from the defining conditional expectation, independence, recurrence, generator, or limiting property; isolate the step where the stochastic assumption is used; then simplify to Model choice checklist: information, state, drift, diffusion, boundary, payoff, measure, numerical scheme, validation. Condition check: state what must be finite, independent, Markov, adapted, stationary, or stable before the result applies. Counterexample pressure: if the learner tries using Black-Scholes or GBM as a universal default instead of matching data and assumptions, construct a small example where the conclusion fails. Repair move: force a model-choice justification before any formula and list at least one failure mode Exam variant: Choose between GBM, OU, reflected diffusion, or a jump model for four short scenarios and justify one limitation of each choice.

Readiness Benchmark Pack

Readiness means the learner can solve closed-note, time-limited stochastic-calculus tasks across Brownian motion, filtrations, quadratic variation, Ito integrals, Ito formula, SDEs, geometric Brownian motion, Ornstein-Uhlenbeck models, generators, Feynman-Kac, Girsanov, Black-Scholes, stopping, and Euler-Maruyama.

Original Timed Mock Final Blueprint

Closed-note practice. Suggested duration: 150 minutes. Show assumptions, method choice, calculations, and interpretation.

1. Task family: adaptedness and filtration checks. Solve one original problem and include a two-sentence model-choice justification.
2. Task family: quadratic variation explanation. Solve one original problem and include a two-sentence model-choice justification.
3. Task family: Ito isometry moment calculation. Solve one original problem and include a two-sentence model-choice justification.
4. Task family: Ito formula derivation. Solve one original problem and include a two-sentence model-choice justification.
5. Task family: SDE integral-form reading. Solve one original problem and include a two-sentence model-choice justification.
6. Task family: geometric Brownian motion and OU solution. Solve one original problem and include a two-sentence model-choice justification.
7. Task family: generator and Feynman-Kac PDE setup. Solve one original problem and include a two-sentence model-choice justification.
8. Task family: Girsanov drift-shift calculation. Solve one original problem and include a two-sentence model-choice justification.
9. Task family: Black-Scholes PDE and hedge derivation. Solve one original problem and include a two-sentence model-choice justification.
10. Task family: barrier hitting and optional sampling. Solve one original problem and include a two-sentence model-choice justification.
11. Task family: Euler-Maruyama simulation update. Solve one original problem and include a two-sentence model-choice justification.
12. Task family: application model-choice memo. Solve one original problem and include a two-sentence model-choice justification.

Scoring Rubric

• 30 percent setup and assumptions.
• 35 percent correct calculation or derivation.
• 20 percent interpretation and units or model meaning.
• 15 percent trap awareness, limitation, and repair.

Readiness Retake Rule

If a learner misses two tasks in the same family, return to the matching chapter, complete the repair prompt, solve a changed-surface problem, and redo only that family before taking another full mock.