# Covariance of Brownian Motion

What is the covariance function for $U(t)$ if $U(t) = e^{-t}B(e^{2t})$ for $t \geq 0$ where $B(t)$ is standard Brownian motion? Any help would be great

-
The process $U$ is a standard OU process. – Shai Covo Apr 27 '11 at 23:17
See also the answer in math.stackexchange.com/questions/30817/…. – Shai Covo Apr 27 '11 at 23:27
Do you know that ${\rm Cov}(B(s),B(t))=s$ for $0 \leq s \leq t$? – Shai Covo Apr 27 '11 at 23:42

## 1 Answer

Hint: consider $\min\{e^{2s},e^{2t}\}$ for $0 \leq s \leq t$.

-