Expected product of Brownian motion at two times

Question

For a standard Brownian motion $W$ and times $s,t\ge0$ , compute $\mathbb{E}[W_sW_t]$ .

Short answer

$\min(s,t)$ . Take $s\le t$ and write $W_t=W_s+(W_t-W_s)$ . The increment is independent of $W_s$ with mean zero, so $\mathbb{E}[W_sW_t]=\mathbb{E}[W_s^2]=s$ .

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q3.50, p.85; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

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

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