Past exam of the random matrix theory course of the University of Oxford / 2025 / 3 / c / i / Solution

The characteristic function of $B(t)\sim N(0,t)$ gives $\mathbb{E}{e}^{2iB(t)}=e^{-2t}$. Taking real parts and using ${\sin }^{2}b=(1-\cos 2b)/2$,There is also a direct check from the stochastic differential equation. Its Itô integral has zero expectation because its integrand is bounded. Thus $m(t)=\mathbb{E}\Delta (t)$ satisfies $m^{\prime }(t)=1-2m(t)$ and $m(0)=0$, whose solution is the same expression.