Past exam of the probability, measure and martingales course of the University of Oxford / 2021 / 1 / b / ii / Solution

Retain the binary random variables $B_j=1_{\{U_j\le X_{j-1}\}}$. Iterating the recursion givesEach binary history determines the full vector $(X_0,\dots ,X_n)$, so its sigma-algebra is finite by the sigma-algebra of a finite-valued random vector. Conversely, because $0<x<1$, the integer $2^n{X}_n-x$ uniquely determines the digits $B_1,\dots ,B_n$; also $B_j=2X_j-X_{j-1}$. ThusAll $2^n$ binary histories have positive probability. By independence of the driving uniform random variables, along a fixed finite history every intermediate state is in $(0,1)$, so each subsequent conditional branch probability, either $X_{j-1}$ or $1-X_{j-1}$, is strictly positive. It follows that ${\mathcal{F}}_n^X$ has $2^n$ nonempty fibres, each an atom of a sigma-algebra, and exactly $2^{2^n}$ elements.

Adaptedness from the preceding part gives ${\mathcal{F}}_n^X\subseteq {\mathcal{F}}_n$. For $n\ge 1$, however, ${\mathcal{F}}_n$ contains the events $\{U_1\le t\}$ for every $t\in [0,1]$. They have distinct probabilities $t$, so there are infinitely many distinct such events. Consequently the finite sigma-algebra ${\mathcal{F}}_n^X$ cannot equal ${\mathcal{F}}_n$. The inclusion is strict for every $n\ge 1$.