Ito's Lemma on Transformed Brownian Motion

This interview question focuses on applying Ito's lemma to simple transformations of Brownian motion, and using that to analyze martingale properties. The setup considers standard Brownian motion and constructs new processes by scaling or taking powers of it, then asks about their distributional properties and whether they remain martingales. Candidates must distinguish between "looks like a martingale" intuition and the precise conditions under which a transformed process actually has zero drift under the filtration generated by Brownian motion. Questions of this type are common for quant research and derivatives roles, where understanding stochastic calculus at a conceptual and computational level is essential.

The problem leans heavily on Ito's lemma for time- and state-dependent transformations, the behavior of moments of Gaussian random variables, and the formal definition of a martingale. An interviewer is checking whether the candidate can correctly identify the drift and diffusion terms of the transformed processes without skipping the Ito correction term. They also watch for comfort with conditioning on the filtration, recognizing when a process with zero expectation fails to be a martingale, and clearly articulating the link between quadratic variation and the extra second-derivative term.

What it tests

Whenever you transform a stochastic process using a function that depends on time and/or the process itself, the resulting process's dynamics are governed by Ito's lemma. Unlike the classical chain rule, Ito's lemma introduces an additional term involving the second derivative with respect to the stochastic variable, reflecting the quadratic variation of Brownian motion. This extra term is crucial: it often generates a nonzero drift even when the original process is a martingale. The presence or absence of this drift term determines whether the transformed process retains the martingale property. The principle holds because stochastic integrals accumulate variance in a way that deterministic calculus does not, so the second derivative term accounts for the 'roughness' of the Brownian path.