Integral of Brownian motion
= Integral of Brownian motion
{c}
{title2=$\int_0^tB_s\,ds$}
For Brownian motion started at $x$, the time integral is a <Gaussian random variable> with
$$
\mathbb E_x\int_0^tB_sds=xt,
\qquad
\operatorname{Var}\left(\int_0^tB_sds\right)=\frac{t^3}{3}.
$$
The variance follows by integrating the covariance kernel $\operatorname{cov}(B_r,B_s)=\min(r,s)$ twice.