Suppose $B_1(t)$ and $B_2(t)$ are two independent, standard Brownian motions. What is the distribution of \begin{align*} G(t) = \int_0^t B_1(\tau)B_2(t-\tau)d\tau \qquad \end{align*} Or, at least an approximation to $\text{Var}(G(t))$?


(1) How about $B_1, B_2$ being any Gaussian process with covariance functions $c_1(t, s)$ and $c_2(t, s)$?

(2) How about \begin{align*} G(t) = \int_0^t B_2(t-\tau)dB_1(\tau) \qquad \end{align*} For either $B_1, B_2$ being Brownian motions or, more generally, Gaussian processes?


1 Answer 1


use $E[B_n(t)]=0$, $E[B_n(t)B_n(t')]=\min(t,t')$: $${\rm Var}\,G(t)=E[G(t)^2]=\int_0^t \int_0^t E[B_1(\tau)B_1(\tau')]E[B_2(t-\tau)B_2(t-\tau')]\,d\tau d\tau'$$ $$=\int_0^t \int_0^t \min(\tau,\tau')\min(t-\tau,t-\tau')\,d\tau d\tau'=t^4/12$$


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