Brownian Motion Variance Integral

This quant interview question is about understanding stochastic calculus in a concrete Brownian motion setting. It focuses on a stochastic integral that looks simple but hides key subtleties of Itô integration and Brownian path behavior. In quant prep, this type of problem sits at the intersection of probability theory and continuous-time models used in derivative pricing.

It trains your mastery of Itô's framework, comfort with stochastic integrals, and your ability to manipulate moments of Brownian-driven quantities. You need to connect distributions of functionals of Brownian motion with their variances and understand how randomness accumulates over time in continuous-time models.

This matters for quant interviews because pricing, hedging, and risk management rely on exactly these tools. Interviewers use it to check that your stochastic calculus is operational, not just theoretical, and that you can handle core quant finance models under time pressure.

What it tests

Stochastic integrals of the form $\int_0^t f(W_s)\,dW_s$ often become tractable by expressing them in terms of functions of the process $W_t$ using Itô's Lemma. The key is that Itô's Lemma systematically relates the differential of a function of a stochastic process to both its drift and diffusion components, allowing us to rewrite complex integrals as combinations of simpler terms. This works because the quadratic variation of Brownian motion introduces a correction term that is not present in classical calculus, making the stochastic integral expressible in terms of the process's moments. For quadratic functions, this typically leads to expressions involving $W_t^2$ and deterministic time terms, whose moments are well-known. The variance or higher moments of such integrals can then be computed using properties of the normal distribution and its moments, reducing the stochastic problem to a deterministic calculation.