Stochastic Calculus Interview Questions: Ito's Lemma, SDEs & Brownian Motion

Who actually gets asked stochastic calculus

Stochastic calculus shows up in a specific slice of the quant interview market: quant researcher roles at pod shops and hedge funds, derivatives-pricing seats at banks, and options-focused desks. If you are interviewing for a trader role versus a researcher role, the difference matters — trader loops lean on probability and mental math, while QR loops at firms like Citadel and Two Sigma will happily ask you to apply Ito's lemma on a whiteboard, especially if your resume mentions derivatives, an MFE, or a stochastic-processes course.

The good news: interviewers don't ask you to reprove existence and uniqueness of SDE solutions. They test whether you can compute with five tools, and whether you understand what the symbols mean. Most questions are 5–15 minute calculations built from the toolkit below.

The core toolkit

1. Brownian motion properties. $W_0 = 0$; increments are independent and stationary with $W_t - W_s \sim N(0,\, t-s)$; paths are continuous but nowhere differentiable. Know the covariance $\text{Cov}(W_s, W_t) = \min(s,t)$ cold — it is the single most common warm-up.

2. The multiplication rules. $(dW_t)^2 = dt$, $\,dW_t\,dt = 0$, $\,(dt)^2 = 0$. Everything unusual about stochastic calculus flows from the first identity: quadratic variation of Brownian motion over $[0,t]$ is exactly $t$.

3. Ito's lemma. For $f(t, X_t)$ where $dX_t = \mu\,dt + \sigma\,dW_t$:

$$df = \left(\frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2 \frac{\partial^2 f}{\partial x^2}\right)dt + \sigma \frac{\partial f}{\partial x}\,dW_t$$

4. The martingale test. An Ito process is a martingale if and only if its drift term is identically zero (given integrability). This converts "is $X_t$ a martingale?" questions — a staple we cover in depth in the martingale interview guide — into a mechanical Ito computation.

5. The named SDEs. You should recognize these on sight:

Process	SDE	Key fact interviewers probe
Geometric Brownian motion	$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$	Solution $S_t = S_0 e^{(\mu - \sigma^2/2)t + \sigma W_t}$; the $-\sigma^2/2$ correction
Ornstein–Uhlenbeck	$dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t$	Mean reversion; stationary variance $\sigma^2/2\kappa$; model for spreads and rates
Cox–Ingersoll–Ross	$dr_t = \kappa(\theta - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t$	Why the $\sqrt{r_t}$ term keeps rates non-negative

Add Girsanov's theorem for pricing-heavy roles: a change of measure shifts the drift of a Brownian motion but never its volatility. That one sentence answers a surprising number of follow-ups in options pricing interviews.

Worked example: $W_t^2$, martingales, and $\mathbb{E}[W_t^4]$

A genuinely common sequence, asked almost verbatim in QR screens. "Let $X_t = W_t^2$. Find $dX_t$. Is $W_t^2 - t$ a martingale? Now compute $\mathbb{E}[W_t^4]$."

Step 1. Apply Ito's lemma with $f(x) = x^2$, so $f' = 2x$, $f'' = 2$:

$$dX_t = 2W_t\,dW_t + \tfrac{1}{2}\cdot 2\,(dW_t)^2 = 2W_t\,dW_t + dt$$

Step 2. In integral form, $W_t^2 = t + 2\int_0^t W_s\,dW_s$. The Ito integral has zero drift, so $W_t^2 - t = 2\int_0^t W_s\,dW_s$ is a martingale. (This also instantly gives $\mathbb{E}[W_t^2] = t$.)

Step 3. For the fourth moment, apply Ito to $f(x) = x^4$:

$$d(W_t^4) = 4W_t^3\,dW_t + 6W_t^2\,dt$$

Take expectations; the $dW_t$ term vanishes:

$$\frac{d}{dt}\,\mathbb{E}[W_t^4] = 6\,\mathbb{E}[W_t^2] = 6t \quad\Longrightarrow\quad \mathbb{E}[W_t^4] = 3t^2$$

Sanity check: $W_t \sim N(0, t)$ and the Gaussian fourth moment is $3\sigma^4 = 3t^2$. Stating that cross-check out loud is exactly the habit interviewers want to see — the same discipline our expectation problems drill.

The standard follow-up: "Is $W_t^3$ a martingale?" Ito gives $d(W_t^3) = 3W_t^2\,dW_t + 3W_t\,dt$. The drift $3W_t\,dt$ has zero expectation but is not identically zero, so no — $\mathbb{E}[W_t^3] = 0$ for every $t$, yet the process is not a martingale. That distinction is the whole point of the question.

The traps interviewers set

• Dropping the second-order term. Applying ordinary calculus to $\ln S_t$ and losing the $-\sigma^2/2$ drift correction is the classic fail. It is why GBM's median path grows slower than its mean.
• Zero mean $\neq$ martingale. As the $W_t^3$ example shows, you must check the drift term, not just the unconditional expectation.
• The Jensen trap. $\mathbb{E}[e^{\sigma W_t}] = e^{\sigma^2 t/2}$, not $1$. Exponentials of Gaussians pick up a convexity premium — forget it and every lognormal calculation downstream breaks.
• Treating $dW_t/dt$ as a thing. Brownian paths are nowhere differentiable; if your manipulation implicitly divides by $dt$ inside a stochastic term, you have left the rails.
• Girsanov confusion. Under the risk-neutral measure the drift becomes $r$, but $\sigma$ is untouched. Claiming volatility changes under a measure change is an instant red flag.

Discrete intuition transfers well here: many continuous-time questions are limits of random walk problems, and interviewers often let you reason discretely first. For theory depth, Shreve's two volumes remain the canonical reference — our Shreve study guide maps which chapters actually matter for interviews (most of Volume II, chapters 3–5, and very little else).

How to practice

Reading derivations is not the same as producing them under pressure. Work timed problems until Ito applications take under three minutes: start with our stochastic processes question bank, layer in the options pricing bank if your target role prices derivatives, and keep the key identities on one page with the full problem bank for mixed review. Around 400 of QuantVault's 2,800+ problems are free, so you can gauge the difficulty bar before committing to anything.

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Frequently asked questions

Do quant trader interviews ask stochastic calculus?

Usually not in depth. Trader interviews at market makers focus on probability, expected value, and mental math, while stochastic calculus is concentrated in quant researcher and derivatives-pricing interviews. If your resume lists an MFE, a PhD, or a stochastic processes course, expect at least one Ito's lemma question regardless of role.

What is the most common stochastic calculus interview question?

Applying Ito's lemma to a simple function of Brownian motion, such as showing that W_t^2 - t is a martingale or deriving the solution to geometric Brownian motion. The GBM derivation is popular because it tests the (dW)^2 = dt rule and the -sigma^2/2 drift correction in one question.

How much of Shreve do I need for interviews?

Far less than the full two volumes. Volume II chapters 3 through 5 cover Brownian motion, Ito calculus, and risk-neutral pricing, which is where nearly all interview questions live. Girsanov's theorem and the martingale representation theorem are worth knowing as statements, but interviewers rarely ask for their proofs.

Is W_t^3 a martingale?

No. Ito's lemma gives d(W_t^3) = 3W_t^2 dW_t + 3W_t dt, and the drift term 3W_t dt is not identically zero, so the process fails the martingale test. Its unconditional expectation is zero for all t, which is exactly why interviewers use it to check whether candidates confuse zero mean with the martingale property.