Equality in distribution of a ratio of independent RVs

Question

I struggle with the conclusion bit of the following problem (an extension to this and this)

Let $M$ be a positive continuous martingale converging a.s. to zero as $t \to \infty$. Let $M^* := \sup_{t \geq 0} M_t$. For $x > 0$, prove that $\mathbb{P}\left[M^* \geq x |\mathcal{F}_0\right] = 1 \land \frac{M_0}{x}$.

Conclude that $M^*$ has the same distribution as $M_0/U$, where $U$ is independent of $M_0$ and uniformly distributed on $[0, 1]$.

If $M_0$ were discrete, I could use the Monotone Convergence Theorem for conditional expectation to make a statement about $\mathbb{P}\left[M_0/U \geq x | \mathcal{F}_0 \right]$, but right now, not even knowing the distribution of $M_0$, I don't know where to start.

Any ideas/hints?

Answer 1

I'm not sure how usual it is to leave a hint on a three year old question, but I do this for the benefit of people who might stumble upon this question.

Let $U$ be a uniformly distributed random variable on $[0, 1]$ that is independent of $M_0$. Then we have for $z > 0$:

$$\Pr \left(\frac {M_0} U \le z\right) = \int_0^1 \Pr \left(\frac {M_0} z \le x\right) dx$$ Write the probability inside the integral as an expectation and apply Fubini. You should just be able to compare this with the CDF of $M^\ast$ you basically already have.