I am working on Exercise 3.28 of Jean-François Le Gall's Brownian Motion, Martingales and Stochastic Calculus. I want to prove the following fact (which is part of Question 3):
Let $(\mathcal{F}_t)_{t\geq 0}$ be a complete filtration and $B$ be an $(\mathcal{F}_t)$-Brownian motion. Let $\mu\in\mathbb{R}$ and $M_t=\exp\left(\mu B_t-\mu^2 t/2\right)$. The stopped martingale $(M_{t\land\sigma_a})_{t\geq 0}$ is closed if and only if $\mathbb{E}M_{\sigma_a}=1$ where $\sigma_a$ is the stopping time $\inf\left\{t\geq 0: B_t\leq t-a\right\}$, with $a>0$.
The implication is trivial. For the converse, by the optional stopping theorem for nonnegative supermatingales, $$\forall t\geq 0, \text{ a.s.}, \quad M_{\sigma\land t}\geq\mathbb{E}\left[M_{\sigma_a}\mid\mathcal{F}_{\sigma_a\land t}\right]\equiv Z_t$$ Reasoning by contradiction, I am able to show that $$\forall t\geq 0, \text{ a.s.}, \quad M_{\sigma_a\land t}=Z_t$$ So $(Z_t)_{t\geq 0}$ is a modification of $\left(M_{\sigma_a\land t}\right)_{t\geq 0}$. My questions are the following
- To show that the stopped martingale is closed, this is not enough, correct? I need to show that the two processes are indistinguishable?
- I know that if both processes are continuous, then they are indistinguishable. However, I don't know how to show continuity...
Do you have ideas on how to do this (or, if this is the wrong method, how to tackle Question 3)?