Brownian Motion Interview Questions: Increments, Covariance, and Ito Corrections

Suppose a Brownian path moves from time 0 to 3. Are the changes over [0,2] and [1,3] independent? What is Cov(B_1, B_3)? Why does differentiating B_t² as if it were an ordinary smooth function miss a term? These three questions test one idea from different angles: Brownian increments are simple locally, but overlapping time intervals and accumulated squared changes cannot be ignored.

This guide treats standard Brownian motion as a mathematical model and uses university MIT quadratic-variation notes and Oxford's Itô/Stratonovich explanation as technical sources. The timed-interval worksheet and derivations are original practice, not a named employer's interview report. For a distinct market-strategy application, see Analyze profits under random walk and Brownian motion.

Begin with the exact Brownian model

Standard Brownian motion B_t starts at zero, has continuous paths, and has independent Gaussian increments: for 0 ≤ s < t, B_t - B_s is normal with mean zero and variance t-s, and disjoint time increments are independent. The specification is about the probability law, not a claim that any observed price or sensor is truly Brownian. The covariance kernel is Cov(B_s,B_t)=min(s,t), as set out in Oxford's Brownian-motion discussion. Those properties let you derive many interview answers without memorizing a separate formula for every interval.

Say what “independent” modifies. B_1 and B_3 are not independent; the later value contains the earlier path plus a future increment. The increment B_3-B_1 is independent of the past up to time 1 under the standard model. A sample path's segments are not a collection of independent positions. This distinction is the article's core thesis: break expressions into nonoverlapping increments and mark shared time before computing covariance or an Itô correction.

Brownian time must be in consistent units. Rescaling time from seconds to minutes changes increment variance unless a diffusion coefficient is adjusted. A process X_t = μt + σB_t has increment mean μ(t-s) and variance σ²(t-s). Calling it “standard Brownian motion” would hide drift and scale. In an interview, confirm whether B means standard Brownian motion, a drifted process, or a geometric price model before applying min(s,t) mechanically.

Question 1: which increments are independent?

Draw a timeline with marks at 0, 1, 2, and 3. Define A=B_2-B_0, C=B_3-B_1, D=B_2-B_1, and E=B_3-B_2. A and C overlap on (1,2], so they are not independent. D and E cover disjoint intervals, so they are independent. The endpoint 2 being shared is not shared duration and does not create covariance. The interval view is far less error-prone than inspecting subscripts as strings.

Pair	Time supports	Covariance under standard B	Independent?
B_2-B_0, B_3-B_1	(0,2] and (1,3]	1, the overlap length	No
B_2-B_1, B_3-B_2	(1,2] and (2,3]	0	Yes
B_1, B_3-B_1	(0,1] and (1,3]	0	Yes
B_1, B_3	(0,1] and (0,3]	1	No

For the first row, write B_3-B_1 = (B_2-B_1)+(B_3-B_2) and B_2-B_0=(B_1-B_0)+(B_2-B_1). All cross-products between disjoint, centered increments have expectation zero; the shared (1,2] increment has variance 1. Therefore covariance is 1. The table is a reusable worksheet: for a standard process, covariance between two increments with positive-length overlap equals the length of their intersection. That is an inference from independent increments and variance-by-duration, not an additional assumption.

Be precise about the zero-covariance shortcut. Arbitrary zero-covariance variables need not be independent. Brownian increments in this setting are jointly Gaussian, so uncorrelated disjoint increments are independent; more directly, their independence is part of the process definition. If an interviewer changes the process to a non-Gaussian one with correlated or dependent shocks, the table's “independent” column must be re-established rather than carried over by habit.

Question 2: derive Cov(B_s,B_t) and a conditional prediction

For s ≤ t, write B_t = B_s + (B_t-B_s). The second term is independent of B_s, and both have mean zero. Thus Cov(B_s,B_t)=Var(B_s)=s. Swap the labels when t < s; the compact answer is min(s,t). At s=1,t=3, covariance is 1, variances are 1 and 3, and correlation is 1/√3, not 1. Shared early randomness creates dependence without making the values identical.

Now ask what can be predicted at time 1. Given B_1, the future increment to time 3 has mean zero and variance 2, so E[B_3 | B_1]=B_1 and Var(B_3 | B_1)=2. This is a statement about the model conditional on current information, not a guarantee that a traded asset has zero expected return. If the candidate instead conditions on B_3=b and asks about B_1, that is a Brownian-bridge question: joint-Gaussian conditioning gives E[B_1 | B_3=b]=b/3 and conditional variance 1 - 1²/3 = 2/3. The two conditional problems reverse which time is observed; confusing them changes the answer.

For an interview check, take B_1=2: the expected B_3 is 2 with uncertainty variance 2. Take B_3=3: the expected B_1 is 1 under a bridge. These numbers are original teaching examples. State whether the information is available at time 1 or revealed only later. A backtest that uses the bridge estimate as a real-time forecast has looked into the future. That data-leakage analogy helps make the distinction memorable, but the math follows from conditional Gaussian covariance.

Question 3: why does the ordinary chain rule fail for B_t²?

For a smooth deterministic function x(t), the first-order chain rule suggests d(x²)=2x dx. Brownian paths are continuous but sufficiently rough that the accumulated squares of fine increments do not vanish. The MIT lecture on quadratic variation presents the limiting sum of squared Brownian increments over a refining partition: over [0,T] it tends to T in the relevant probabilistic sense. This is the source of the informal mnemonic (dB_t)²=dt.

Use a finite partition to see the missing term. Every increment satisfies (B_{t+Δ}-B_t)² = 2B_t ΔB + (ΔB)² after rearranging B_{t+Δ}²-B_t². Sum across the partition. The left side telescopes to B_T²-B_0²; the last term's sum does not disappear as the intervals shrink. The Itô result is B_T² = 2∫₀ᵀ B_t dB_t + T, or in differential shorthand d(B_t²)=2B_t dB_t+dt. A strong answer points to quadratic variation before invoking a rule by name.

One consistency check is expectation. E[B_T²]=Var(B_T)=T. The adapted Itô integral ∫₀ᵀ B_t dB_t has mean zero under its usual integrability conditions, leaving exactly T. If someone drops the correction term, their expression would incorrectly suggest a zero expected square. The check is not a full proof of Itô's formula, but it quickly catches a sign or factor mistake.

Question 4: apply Itô's formula to a transformed process

For a sufficiently smooth f(t,x), the one-dimensional Itô rule for X_t satisfying dX_t=a(t,X_t)dt+b(t,X_t)dB_t is df(t,X_t)=[f_t+a f_x+(1/2)b² f_xx]dt+b f_x dB_t. The extra second-derivative term is tied to the quadratic variation of the noisy part. Oxford's Itô versus Stratonovich guide explains why the interpretation of a stochastic integral affects the chain rule and drift conversion. Specify the Itô convention before writing these formulas.

With X_t=B_t and f(x)=e^x, the rule gives d(e^{B_t})=e^{B_t}dB_t+(1/2)e^{B_t}dt. With f(x)=x², it gives the prior dt correction. These are not decorative calculus exercises: they show that a nonlinear transformation of a zero-drift Brownian state can acquire a positive drift in expectation. Calling the transformed variable a martingale merely because B has zero mean would be wrong. Test your expression against a known moment, such as E[e^{B_t}]=e^{t/2} for standard Brownian motion.

For a geometric model dS_t=μ S_t dt+σ S_t dB_t with positive S_t, apply f(S)=log S. Derivatives are 1/S and -1/S², so d log S_t=(μ-σ²/2)dt+σ dB_t. The -σ²/2 is the Itô correction. It does not mean the expected simple price growth rate is automatically μ-σ²/2; expected log growth and expected level growth are different quantities. State which expectation the interviewer asks for, and do not interpret the model as a guarantee about real market prices.

Question 5: how would you simulate and test the model?

On a grid of width Δt=T/n, draw independent standard-normal values Z_k and set ΔB_k=√Δt Z_k. Accumulate the increments from zero. The square root is essential: variance of each step is Δt, not (Δt)². For repeated simulated paths, check that a selected increment has approximately zero mean and variance equal to its interval length, and that disjoint increments have approximately zero sample covariance. A single path cannot establish distributional properties; use many independent paths and acknowledge Monte Carlo error.

Add a quadratic-variation check: V_n=Σ(ΔB_k)². Its expected value is T because each squared increment has mean Δt. For normal increments, Var(V_n)=2T²/n, so its dispersion shrinks as the mesh refines. At T=1,n=100, mean is 1 and variance is 0.02 across paths; no individual path is required to equal exactly 1. Compare this with ordinary signed-increment sums: ΣΔB_k=B_T, whose variance stays T, so it does not collapse to a deterministic total.

For Itô discretization, use left-endpoint integrands: Σ B_(t_k) ΔB_k approximates ∫ B_t dB_t. A midpoint rule corresponds to a different stochastic integral convention and changes the correction relationship. If a numerical experiment “disproves” B_T²=2∫ B dB+T, inspect the evaluation point and time scaling before questioning the theorem. The Oxford explanation discusses precisely this convention distinction.

Question 6: what fails when the model is used as a shortcut?

An interview may move from mathematical Brownian motion to observed returns, physical trajectories, or trading rules. Separate the mathematical derivation from empirical adequacy. Real processes can have drift, changing volatility, jumps, autocorrelation, bounded states, microstructure noise, or observation gaps. The Brownian formulas remain correct under their own assumptions, but the inference from those formulas to a product or market claim requires validation. If interval variances do not scale linearly with elapsed time, or disjoint increments show dependence, investigate the mismatch instead of relabeling the data “approximately Brownian” without a test.

Likewise, a trading rule that sees an increment and then acts after it cannot earn that past increment retroactively. Write the observation time, entry time, exit time, and units of exposure before calculating profit. Preserve that scope distinction. A candidate who can derive overlap covariance but ignores when a strategy becomes measurable still has an incomplete answer.

An interview-ready response is short: “I draw the time supports, decompose overlapping increments, and get covariance from shared duration. I derive min(s,t) from independent future increments. For nonlinear functions I include quadratic variation, so Itô's formula adds a half-second-derivative term. I test a simulation against variance-by-time and Σ(ΔB)² → T, then check whether the application satisfies the Brownian assumptions.” Each clause has a derivation behind it.

Sources and further reading

• MIT OpenCourseWare: Quadratic variation — squared-increment limit underlying the Itô correction.
• University of Oxford: Itô and Stratonovich, a guide for the perplexed — Brownian covariance, stochastic-integral convention, and chain-rule interpretation.
• MIT: Brownian motion lecture slides — Brownian increments and Itô formula in course material.