Let $W$ be a Wiener process and $X(t):=W^{2}(t)$ for $t\geq 0.$ Calculate $\operatorname{Cov}(X(s), X(t))$.

Question

Let $W$ be a Wiener process. If $X(t):=W^2(t)$ for $t\geq 0$, calculate $\operatorname{Cov}(X(s),X(t))$

Answer 1

Assume WLOG that $t\geqslant s$. The main task is to compute $E(X_sX_t)$.

Write $X_t=(W_t-W_s+W_s)^2=(W_t-W_s)^2+2W_t(W_t-W_s)+W_s^2$, and use the fact that if $U$ and $V$ are independent random variables, so are $U^2$ and $V^2$.