The two-dimensional <Itô formula>, using $d[W^X,W^Z]_t=\rho\,dt$, gives the drift of $U(t,Z_t,X_t)$ as
$$
U_t+BU_z+\frac12C^2U_{zz}
+zC\rho U_{zx}
+\frac12z^2(U_{xx}-U_x).
$$
The PDE makes this zero, leaving only stochastic-integral terms. Thus $M_t=U(t,Z_t,X_t)$ is a <local martingale>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 6 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-211/6/a/solution