Covariance of Wiener Process with nonzero $x(0)$

Question

Let $x(t)$ be a Wiener process and let $\sigma_1^2 = var(x(1))$, $\sigma_0^2 = var(x(0))$. If $x(0) = 0$, then we know that $C(t,s) = \sigma_1^2 min(t,s)$ (please see Mean and covariance of Wiener process). I want to prove that if $x(0)$ is not zero and instead a r.v then $C(t,s) = \sigma_0^2 + min(t,s)(\sigma_1^2 - \sigma_0^2)$. Any ideas to start with?

Answer 1

Write $x(t)=B(t)+x(0)$ where $B$ is a Wiener process started at $0$ with Var$[B(1)]=\sigma_1^2-\sigma_0^2$ and $x(0)$ is independent of $B$. Now use bilinearity of covariance to obtain the claim.