Integral of Brownian Motion

This question focuses on the time-integral of a standard Brownian motion, viewed as a continuous-time analogue of summing correlated Gaussian increments. The candidate must reason about what kind of random variable this integral produces, how its distribution relates to the underlying process, and what structural properties (such as scaling and dependence on the time horizon) can be extracted. It sits in the core of stochastic calculus and continuous-time modeling, and often appears in interviews for quantitative research and derivatives roles where understanding functionals of Brownian paths is essential for pricing and risk calculations.

On the technical side, the question leans on recognizing linear functionals of Gaussian processes, working with covariance kernels, and manipulating integrals of random functions. A strong answer uses the joint Gaussianity of Brownian motion, carefully sets up expectations and covariances, and evaluates the required integrals cleanly. Interviewers look for comfort with interchanging integrals and expectations, checking integrability and measurability conditions, and using symmetry or scaling arguments rather than brute-force calculation. Precision in defining the object, identifying its law, and articulating its dependence on the time horizon is more important than heavy algebra.

What it tests

When integrating a Gaussian process like Brownian motion over time, the resulting random variable is again Gaussian because linear operations preserve normality. The mean and variance of this new variable are determined by the mean and covariance structure of the original process. Specifically, the mean of the integral is the integral of the mean function, and the variance is a double integral over the covariance kernel. This pattern holds for any process with jointly Gaussian increments and a known covariance, allowing us to translate properties of the process into properties of its time-integral. The key is that the integral essentially 'sums up' the correlations across time, amplifying the variance in a way that reflects the process's memory and dependence structure.