Since $V'=v$, the <fundamental theorem of calculus> gives
$$
\psi'(x)=e^{-V(x)},
\qquad
\psi''(x)=-v(x)e^{-V(x)}.
$$
Applying the <Itô formula> before the lifetime, the two drift terms cancel:
$$
d\psi(X_t)
=\frac12v(X_t)\psi'(X_t)dt
+\psi'(X_t)dB_t
+\frac12\psi''(X_t)dt
=e^{-V(X_t)}dB_t.
$$
Thus $\psi(X)$ is a continuous local martingale. The increasing function $\psi$ is the <scale function of a one-dimensional diffusion>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 5 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-202/5/b/solution