Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 4

Continuity gives $X_{{\tau }_a}=a$ on $\{{\tau }_a<\infty \}$. The stopped process $X^{{\tau }_a}$ is bounded by $a$ and is therefore a true martingale. HenceThe second term tends to zero by bounded convergence because $X_t\to 0$ and it is bounded by $a$. Thus

The Dambis-Dubins-Schwarz theorem says that there is Brownian motion $W$ such thatWhen $[M]$ is strictly increasing, define its inverseand set $W_s=M_{T_s}$. Optional sampling shows that $W$ is a continuous local martingale, while time change gives $[W{]}_s=s$. The Lévy characterization of Brownian motion makes $W$ Brownian, and inverse time change gives the displayed representation.

Set $s=[M{]}_t$. By hypothesis $s\to \infty$, and part b givesSince $W_s/s\to 0$ almost surely, the expression tends to $-\infty$ almost surely.

The Doléans-Dade exponentialis a positive continuous local martingale with $X_0=1$, and part c shows $X_t\to 0$. Apply part a with $a=e^y$: