Why does $\langle W(t) W(s) \rangle = \min (t,s)$ for Wiener process $W$?

Question

In section 4.2 of Gardiner's Stochastic Methods, we find

\begin{align} & \left\langle \sum_{i=1}^n W(\tau_i)[W(t_i) - W(t_{i-1})]\right\rangle \\[10pt] = & \sum_{i=1}^n [ \min(\tau_i,t_i) - \min(\tau_i,t_{i-1})] \end{align}

and I do not understand this step, although I do understand that $\langle W(t)^2 \rangle = t$.

Answer 1

Suppose for simplicity that $s<t$. Then using independence of increments, we get $$ \mathbb{E}[W_tW_s]=\mathbb{E}[(W_t-W_s)W_s]+\mathbb{E}[W_s^2]=\mathbb{E}[W_t-W_s]\mathbb{E}[W_s]+\mathbb{E}[W_s^2]=s$$