Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 30 / 2 / b

Write $X_t=x+W_t+\mu t$ with $W$ a zero-starting Brownian motion. A unit-time increment exceeds $2b$ with probabilityIf the process has not exited before an integer time $k$, it is in $(-b,b)$; an increment exceeding $2b$ forces it outside at time $k+1$ and hence forces an exit by then. Independence of Brownian increments gives $\mathbb{P}(T>k)\le (1-p{)}^k$. Thus the geometric tail bound from a uniform escape probability yieldsIn particular $T$ is finite almost surely.

The diffusion generator is $\mathcal{L}=\frac{1}{2}{\partial }_{xx}+\mu {\partial }_x$. The function $f(z)=e^{-2\mu z}$ satisfies $\mathcal{L}f=0$, so the Itô formula gives $df(X_t)=-2\mu f(X_t)d{W}_t$. It is a true martingale, since it is $e^{-2\mu x}$ times the Gaussian exponential martingale $e^{-2\mu W_t-2{\mu }^{2}t}$. The stopped values $f(X_{t\wedge T})$ are uniformly bounded because the stopped path stays in $[-b,b]$. Optional sampling and bounded convergence givewhere continuity ensures that $X_T$ is exactly one of the two endpoints. Solving,This drifted Brownian interval-exit probability tends to $(b-x)/(2b)$ as $\mu \to 0$, as expected.