For continuous local martingales $M$ and $N$, the <Itô product rule> says that
$$
M_tN_t-M_0N_0-[M,N]_t
$$
is a <local martingale>. If the martingales are square-integrable and converge in $L^2$, then
$$
\mathbb E[M_\infty N_\infty]=\mathbb E[M_0N_0]+\mathbb E[M,N]_\infty.
$$

Martingale product identity

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

{c}

For continuous semimartingales,
$$
d(X_tY_t)=X_t\,dY_t+Y_t\,dX_t+d[X,Y]_t.
$$
It is the stochastic counterpart of the ordinary <product rule>; the <quadratic covariation> supplies the additional second-order term.

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