Stochastic Integral: w(t) dw(t)

This question focuses on a basic but subtle stochastic integral where Brownian motion is integrated with respect to itself over a fixed time interval. The candidate is asked to reason about how such an object should be interpreted in the Itô sense, and what can be said about its distributional properties, expectation, and pathwise behavior. Because of its centrality to pricing and hedging arguments, variants of this setup often appear in quant finance interviews, especially for roles involving stochastic calculus, derivatives modeling, or quantitative research in banks and hedge funds.

To handle this integral correctly, the candidate must apply Itô's Lemma to a suitably chosen function of Brownian motion and carefully track the quadratic variation term. The question leans on understanding the difference between Itô and classical Riemann–Stieltjes integrals, recognizing martingale structures, and relating the integral to known processes or functionals of Brownian motion. Interviewers watch for comfort manipulating differentials, clarity on why the extra correction term appears, and the ability to connect the final expression to properties such as mean, variance, and martingale behavior without hand-waving.

What it tests

In stochastic calculus, integrals of the form $\int_0^T f(w(t), t) dw(t)$ are governed by Itô's Lemma, which generalizes the chain rule for stochastic processes. The key insight is that the differential of a function of a stochastic process includes not only the usual derivative term but also a correction term involving the process's quadratic variation. This correction arises because the increments of Brownian motion have nonzero variance even over infinitesimal intervals, so $dw(t)^2$ is of order $dt$. As a result, when integrating, the stochastic integral is not simply the antiderivative as in classical calculus; it must account for this extra term, fundamentally changing the relationship between the process and its integral. This structure underpins the martingale property of Itô integrals and explains why expectations behave differently than in deterministic calculus.