Law of the time integral of Brownian motion

Question

Let $w$ be a standard Brownian motion. What is the distribution of the random variable $\int_0^T w(t)\,dt$ ?

Short answer

Normal with mean $0$ and variance $T^3/3$ . It is a limit of linear combinations of Gaussians, so it is Gaussian with mean $0$ . Its variance is $\int_0^T\int_0^T\min(s,t)\,ds\,dt=T^3/3$ . Equivalently, integrating by parts gives $\int_0^T(T-t)\,dw(t)$ , whose Ito-isometry variance is again $T^3/3$ .

Adapted from Crack, Heard on the Street, Q2.29, p.28; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

The Four Faces of Brownian Motion
Covariance and Correlation
Itô's Formula

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free