25 original stochastic calculus interview questions with explained answers on Brownian motion, Itô integrals, SDEs, martingales and measure changes.
The extra drift term in Itô calculus changes expected growth, option pricing and hedge dynamics. Begin with Brownian increments and quadratic variation, then derive simple stochastic integrals and diffusion moments. Throughout this guide, Brownian motion is standard and integrals are Itô integrals unless stated otherwise. Integrability and stopping conditions are part of the answer, especially when reasoning about martingales.
For dX = a dt + b dW, the quadratic term is b²dt. In more than one dimension, include the cross variation of correlated drivers.
A deterministic-integrand Brownian integral is Gaussian; a random-integrand integral need not be. Isometry computes a second moment without establishing a full distribution.
A zero formal drift may establish only a local martingale. Verify integrability before taking expectations, changing measure or invoking optional stopping.
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Normal increments coexist with irregular paths. Quadratic variation explains why ordinary differentiation is insufficient.
For standard Brownian motion W, what distribution does Wₜ − Wₛ have for t > s?
Answer: Normal with mean zero and variance t − s.
Explanation: Brownian increments scale with elapsed time in variance. Their standard deviation is √(t − s), not t − s.
Find Cov(Wₛ,Wₜ) for standard Brownian motion.
Answer: min(s,t).
Explanation: For s ≤ t, write Wₜ = Wₛ + an independent increment. Only the shared Wₛ contributes covariance.
What is the quadratic variation of standard Brownian motion over [0,T]?
Answer: T, almost surely.
Explanation: Sums of squared increments converge despite shrinking individual increments. This nonzero quadratic variation produces Itô's correction term.
Does a standard Brownian path have finite total variation on a positive finite interval?
Answer: No, almost surely.
Explanation: Its paths are continuous but too irregular for ordinary finite-variation calculus. Continuity alone does not justify a Riemann–Stieltjes integral against Brownian motion.
Find d(Wₜ²).
Answer: 2Wₜ dWₜ + dt.
Explanation: The second derivative of x² is 2. Multiplying by half the Brownian quadratic variation adds dt.
Differentiate the function, include the correction term and integrate only after checking the assumptions.
Is Wₜ² − t a martingale?
Answer: Yes, for standard Brownian motion with its usual filtration.
Explanation: Itô's formula leaves differential 2Wₜ dWₜ, and finite-time integrability gives a true martingale here.
For constant a, what compensator makes exp(aWₜ) a martingale?
Answer: exp(aWₜ − a²t/2).
Explanation: Itô's formula cancels the drift with the deterministic correction. For constant a, Gaussian exponential moments justify the martingale property.
Find d(Wₜ³).
Answer: 3Wₜ² dWₜ + 3Wₜ dt.
Explanation: The half-second-derivative term is (1/2) × 6Wₜ dt. Ordinary differentiation would omit it.
Express the Itô integral from 0 to T of Wₜ dWₜ.
Answer: (W_T² − T)/2.
Explanation: Integrate the differential of W² and rearrange. The −T term distinguishes Itô integration from ordinary smooth-path calculus.
For an adapted square-integrable H, express E[(∫₀ᵀ Hₜ dWₜ)²].
Answer: E[∫₀ᵀ Hₜ² dt].
Explanation: The isometry turns a stochastic second moment into an ordinary time integral. Adaptedness and square integrability are essential assumptions.
Isometry calculates uncertainty while correlation creates mixed variation terms.
Find the mean and variance of ∫₀ᵀ t dWₜ.
Answer: Mean zero; variance T³/3.
Explanation: A deterministic square-integrable integrand has zero Itô mean. Isometry gives integral of t²; the integral is also Gaussian.
Find the variance of ∫₀ᵀ Wₜ dWₜ.
Answer: T²/2.
Explanation: The mean is zero and isometry gives ∫₀ᵀ E[Wₜ²]dt = ∫₀ᵀ t dt. The integral is not generally Gaussian.
Brownian motions W and B have constant instantaneous correlation ρ. What is dWₜ dBₜ in Itô notation?
Answer: ρ dt.
Explanation: Cross variation contributes mixed second-derivative terms in multivariate Itô formulas. Independent drivers have ρ = 0.
If dX = a dt + b dW, state df(t,X) for a sufficiently smooth f.
Answer: (f_t + a f_x + b²f_xx/2)dt + b f_x dW.
Explanation: The drift combines explicit time change, ordinary first-order movement and quadratic variation. Evaluate derivatives at the current state.
Solve dS = μS dt + σS dW with positive S₀ and constant coefficients.
Answer: Sₜ = S₀ exp[(μ − σ²/2)t + σWₜ].
Explanation: Apply Itô to log S, integrate, then exponentiate. Omitting the variance correction produces the wrong mean.
Log transformations and integrating factors give useful closed forms for standard diffusion models.
For that geometric Brownian motion, find E[Sₜ].
Answer: S₀ exp(μt).
Explanation: The Gaussian exponential moment exp(σ²t/2) cancels the correction in log drift. Expected price and exponentiated expected log price differ.
For that GBM, find Var(Sₜ).
Answer: S₀² exp(2μt)[exp(σ²t) − 1].
Explanation: Compute the second moment using the normal moment-generating function, then subtract the squared first moment.
If dX = κ(θ − X)dt + σdW with κ > 0 and fixed X₀ = x, find E[Xₜ].
Answer: θ + (x − θ)exp(−κt).
Explanation: Taking expectations yields a linear mean-reversion ODE. The mean tends to θ at a rate governed by κ.
For that OU process with deterministic initial state, find Var(Xₜ).
Answer: σ²[1 − exp(−2κt)]/(2κ).
Explanation: Use the integrating-factor solution and Itô isometry. As t grows, variance approaches σ²/(2κ).
For dX = a(X)dt + b(X)dW, what is the generator acting on a smooth f(x)?
Answer: Lf = a f′ + b²f″/2.
Explanation: The generator is the drift term from Itô's formula for time-independent f. It links diffusion dynamics to backward equations.
Track the time scaling, measure and integrability conditions behind a numerical or pricing argument.
Give one Euler–Maruyama step of width Δt for dX = a(X)dt + b(X)dW.
Answer: X_next = X + a(X)Δt + b(X)√Δt Z, with Z standard normal.
Explanation: Brownian increments have variance Δt. Using Δt instead of √Δt on the noise term changes the simulated diffusion.
With constant risk-free rate r and continuous dividend yield q, what is the stock drift under a standard risk-neutral GBM model?
Answer: r − q.
Explanation: The discounted total-return process is a martingale under the pricing measure. Include dividends when translating this into ex-dividend price dynamics.
Let Z_T = exp(−θW_T − θ²T/2) define Q by dQ/dP = Z_T. Which shifted process is Q-Brownian on [0,T]?
Answer: Wₜ + θt.
Explanation: The sign follows from the negative sign in the density exponent. Constant θ satisfies the usual exponential-integrability condition on a finite horizon.
Is every nonnegative local martingale necessarily a true martingale?
Answer: No; it is a supermartingale and may be a strict local martingale.
Explanation: Localisation does not by itself preserve expectation globally. Additional integrability is needed to make a true-martingale claim.
Can you always replace a deterministic time by an arbitrary stopping time in a martingale expectation identity?
Answer: No.
Explanation: Optional stopping requires suitable boundedness or integrability conditions. Check them before using an apparently fair process to calculate a stopping payoff.
Each question has its own page with a short answer and a credit to its source. Hints and full worked solutions are in the question bank with a free account.
Our bank summary dated contains 57 Stochastic Calculus entries in the bank. Counts describe entries in our curated practice bank, including its source and classification choices. They do not measure employer interview frequency, candidate success rates or how often any one question is asked.
The public examples above were written for these guides. They illustrate the techniques and are not a reproduction or ranking of the bank's sourced questions. The dashboard's broader bank combines sourced questions, worked answers where available and references for book entries without a reproduced solution.
| Subtopic | Bank entries |
|---|---|
| Brownian Motion | 31 |
| Ito Calculus | 21 |
| SDEs | 3 |
| Change of Measure | 2 |
Bank difficulty labels: 23 easy · 22 medium · 12 hard.
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