Because independent <Brownian motions> have zero <quadratic covariation>, the <Itô product rule> gives
$$
d(B_t^1B_t^2)=B_t^1\,dB_t^2+B_t^2\,dB_t^1.
$$
After integration, the random variable in the question is $B_t^1B_t^2$. Writing $B_t^j=\sqrt t\,Z_j$ for independent standard <Gaussian random variables> $Z_1,Z_2$, its distribution is
$$
tZ_1Z_2,
$$
the scaled <product of two independent standard normal random variables>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-202/1/c/solution