Integral of Brownian motion (${\int }_0^t{B}_{s}ds$)

For Brownian motion started at $x$, the time integral is a Gaussian random variable withThe variance follows by integrating the covariance kernel $\mathrm{cov}(B_r,B_s)=\min (r,s)$ twice.

The displayed continuous centered Gaussian process, with value zero at time zero, has the Brownian covariance kernel. Hence it is a Brownian motion in its own natural filtration. More generally, replacing three by $c$ gives covariance $s+c(c-3)(s/2-s^2/(6t))$ for $0<s\le t$, so three is the only nonzero valid coefficient. Its conditional future increment in the original Brownian filtration is $3(t-s)t^{-1}(s^{-1}{\int }_0^s{W}_{u}du-W_s)$, and is generally nonzero.