I am reading through Le Gall's book on Brownian Motion, Martingales, and Stochastic Calculus. I just read through the chapter on optional stopping of martingales, but I cannot solve the first exercise:

  1. Let $M$ be a martingale on a probability space with a complete filtration such that $M$ has continuous stopping times, and $M_0 = x \in \mathbb{R}_+$. We assume $M_t \geq 0$ for every $t \geq 0$, and that $M_t \to 0$ when $t \to \infty \; $ a.s. Show that, for every $y > x,$ $$P(\sup_{t \geq 0} M_t \geq y) = \frac{x}{y}$$

  2. Give the law of $\sup_{t \leq T_0} B_t$, when $B$ is a Brownian motion started from $x > 0$ and $T_0 = \inf \{t \geq 0 : B_t = 0 \}$

  3. Assume now that $B$ is a Brownian motion started from $0$, and let $\mu > 0$. Using an appropriate exponential martingale, show that $\sup_{t\geq 0}(B_t - \mu t)$ is exponentially distributed with parameter $2\mu$.

Now, I am able to upper bound the probability $P(\sup_{t \geq 0} M_t \geq y) \leq \frac{x}{y}$ by using one of the Doob martingale inequalities. However, I am not sure how to finish off the proof of 1. I am assuming that I would have to use optional stopping, but I cannot make a useful stopping time.

  • $\begingroup$ If your question concers only the first part of the exercise, then why post the other two unrelated parts of the exercise? The first part is solved here $\endgroup$ – saz Apr 15 at 17:28

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