Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 3

For a continuous local martingale $M$ with $M_0=0$, its stochastic exponential isThe Itô formula gives $d\mathcal{E}(M{)}_t=\mathcal{E}(M{)}_{t}d{M}_t$, so $\mathcal{E}(M)$ is a positive continuous local martingale. Every nonnegative local martingale is a supermartingale, because localization and the Conditional Fatou lemma turn the localized martingale equality into the supermartingale inequality.

The exponential process $Z_t=\exp (-B_t-t/2)$ is a martingale. Under the Girsanov theorem change of measure $d\mathbb{Q}{|}_{{\mathcal{F}}_t}=Z_t,d\mathbb{P}{|}_{{\mathcal{F}}_t}$, the process $W_t=B_t+t$ is Brownian motion. Its first hitting time $T_a$ of $a>0$ is finite $\mathbb{Q}$-almost surely. On $\{T_a\le t\}$, the optional sampling theorem givesbecause $B_{T_a}=a-T_a$. Letting $t\to \infty$ and applying the monotone convergence theorem yieldsThis is the Critical exponential moment of a drifted Brownian hitting time.

By the Reflection invariance of Brownian motion, $-B$ is Brownian motion, and the first time $B_t-t$ reaches $b<0$ is the first time $-B_t+t$ reaches $-b>0$. Part b therefore gives

Now $L_t=\exp (B_t-t/2)$ is the exponential Brownian martingale. At $S_b$, the identity $B_{S_b}=b+S_b$ gives $L_{S_b}=e^{b+S_b/2}$, so $\mathbb{E}{L}_{S_b}=1$. The nonnegative stopped martingale $(L_{t\wedge S_b})$ therefore loses no mass at infinity and is uniformly integrable. The optional sampling theorem at any stopping time $R$ gives

Put $A_t=[M{]}_t$ and use the Dambis-Dubins-Schwarz theorem to write $M_t=B_{A_t}$, enlarging the space if necessary after the terminal clock value. In the time-changed filtration, $A_t$ is a stopping time. For $b<0$, let $S_b=\inf \{s:B_s-s=b\}$. Applying part c with $R=A_t$ givesOn $\{S_b\le A_t\}$, the first integrand is at most $e^b{e}^{A_t/2}$. The assumed Novikov condition therefore impliesas $b\to -\infty$, uniformly for $t\le T$. Meanwhile $mathbf{1}_{\{A_t<S_b\}}\uparrow 1$, so the monotone convergence theorem gives $\mathbb{E}\mathcal{E}(M{)}_t=1$. A nonnegative local martingale with constant expectation is a martingale. Thus $\mathcal{E}(M^T)$ is a martingale, proving the Novikov condition.