Weighted Brownian Motion Variance

This quant interview question is about understanding stochastic integrals with Brownian motion in a clean, tractable setting. It focuses on how a deterministic time-dependent weight interacts with continuous-time randomness, a core theme in stochastic calculus and continuous-time finance models. For quant prep, it is a classic warm-up to more complex problems involving Itô processes and diffusion models.

It trains your grasp of how variance behaves for integrals driven by Brownian motion, and how deterministic functions modulate randomness over time. You reinforce intuition about scaling, homogeneity in time, and the way uncertainty accumulates in continuous-time models. It also consolidates your comfort switching between probabilistic reasoning and standard calculus.

This matters in quant interviews because pricing, hedging, and risk modeling often reduce to understanding such integrals. Interviewers use it to check that your stochastic calculus foundations are solid before moving on to SDEs, option pricing, and interest-rate models. For serious quant interviews and quant prep, this type of variance computation is considered basic fluency.

What it tests

For stochastic integrals of the form $X_t = \int_0^t f(s) \, dB_s$ where $f(s)$ is a deterministic function and $B_t$ is standard Brownian motion, the variance of $X_t$ is given by the Itô isometry: $\operatorname{Var}(X_t) = \int_0^t (f(s))^2 \, ds$. This arises because increments of Brownian motion are independent and have variance proportional to the length of the interval, so the quadratic variation accumulates according to the square of the integrand. The key is that the randomness comes entirely from the Brownian motion, and the deterministic function simply scales the contribution of each infinitesimal increment. This structure means that the variance calculation reduces to a deterministic integral, regardless of the complexity of $f(s)$.