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

For $T_n=T\wedge n$, Itô formula and the differential equation show thatup to $T$. After localization this is a martingale, and boundedness of $V$ permits optional stopping. Thus, conditionally on $X_0$,On $\{T<\infty \}$, path continuity gives $X_T\in \{\ell ,r\}$ and hence $V(X_T)=1$. On $\{T=\infty \}$, boundedness of $V$ makes $e^{-\lambda n}V(X_n)\to 0$. The dominated convergence theorem therefore yieldson $\{\ell <X_0<r\}$, with the stated convention when $T=\infty$.