# Continuous-time Martingale and Brownian Motion Supremums

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.

• 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 – saz Apr 15 at 17:28