I have a real number $x$, and $W$ is a standard Brownian motion. Let $0 < s < t$. How to find
$$
\mathsf E[W_s | W_t = x]
$$
Please provide me with a step by step answer as I want to understand your steps and
the concept.
Many thanks in advance.
Best Answer
Let $U_{s,t}=W_s-(s/t)W_t$ and $V_{s,t}=(s/t)W_t$. The process $W$ is centered gaussian, $\mathbb E(W_t^2)=t$ and $\mathbb E(W_sW_t)=s$, hence $U_{s,t}$ and $W_t$ are independent. Thus, $W_s=U_{s,t}+V_{s,t}$ where $U_{s,t}$ is centered and independent of $W_t$ and $V_{s,t}$ is measurable with respect to $W_t$.
Thus, $\mathbb E(W_s\mid W_t)=\mathbb E(U_{s,t}\mid W_t)+\mathbb E(V_{s,t}\mid W_t)=$ $____$ $+$ $____$ $=$ $____$.