# Let $X_t$ be an Ornstein Uhlenbeck process. Is $X_t - \frac{1}{2} \int_o^{t} X_u du$ a Brownian motion?

Let $$X_t=e^{-t}B_{e^{2t}}$$ be an Ornstein Uhlenbeck process with $$\mathbb{E}[X_tX_s]=e^{-|t-s|}$$. Is $$Y_t = X_t - \frac{1}{2} \int_o^{t}X_u du$$ a Brownian motion? I managed to prove that $$Y_t$$ is a centered Gaussian process. The only thing I need to show is that it has $$\mathbb{E}[Y_sY_t]=min(s,t)$$, but I could not figure this out. Any helps would be appreciated.

Hint: $$EX_tX_s=e^{-t-s} (e^{2t}\wedge e^{2s})$$. $$EX_t\int_0^{s}X_u du=\int_0^{s} E(X_tX_u)du$$ and $$E\int_0^{t} X_u du \int_0^{s}X_vdv=\int_0^{t}\int_0^{s} E(X_uX_v)dudv$$. All these are easy to compute.
• It should be $E\int_0^{t} X_u du \int_0^{s}X_vdv=\int_0^{t}\int_0^{s} E(X_uX_v)dudv$. This is still easy to compute but it does not end up with what I expect. – pac Sep 24 at 0:38