The stochastic-integral identity obtained in part (i), evaluated at $s=t$, is
$$
U(0,X_t)=U(t,X_0)+\int_0^t
\partial_xU(t-s,X_s)\sigma(X_s)dW_s.
$$
Since $dX_s-b(X_s)ds=\sigma(X_s)dW_s$, this has the requested form with the <previsible process>
$$
\theta_s=\partial_xU(t-s,X_s).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 4 / d / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-202/4/d/ii/solution