Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 32 / 3 / b / Solution

On a product extension of the probability space, take a uniform distribution variable $U$ on $(0,1)$ independent of the original sigma-algebra. For $x>0$,These are the same conditional tails as in the maximal identity for a continuous nonnegative local martingale tending to zero. Taking expectations identifies the unconditional probability distributions:If $M_0=0$, define $M_0/U=0$; the endpoint $U=0$ is a null event if one uses $[0,1]$ instead. The possible atom at zero has mass $\mathbb{P}(M_0=0)$, since the conditional probability of a positive maximum is zero exactly on that event. For fixed $M_0=m>0$, the conditional law is the Pareto distribution of shape one and minimum $m$, with density $m/x^2$ on $x>m$.