Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 6

Every evaluation map ${\pi }_t$ is continuous in the uniform metric, so $\mathcal{F}\subseteq \mathcal{B}$. Conversely, for $g\in C[0,1]$,with the harmless restriction to $m$ for which $r-1/m>0$. Thus every open ball belongs to $\mathcal{F}$. Separability makes every open set a countable union of balls, so $\mathcal{B}\subseteq \mathcal{F}$ and equality follows.

Let $d(x,A)$ be the distance to the closed set $A$. Continuity givesEvery variable in the countable infimum is ${\mathcal{F}}_t$-measurable by adaptedness, so the event belongs to ${\mathcal{F}}_t$. Hence $T_A$ is a stopping time.

Almost-sure convergence plus uniform integrability gives $X_n\to X$ in $L^1$. Conditional expectation is an $L^1$ contraction, so

Take $X_n=Y_n{Z}_n$ and $\mathcal{G}=\sigma (Y_1,Y_2,\dots )$. Then $\mathbb{P}(X_n=n)=n^{-2}$, so $(X_n)$ is uniformly integrable and $X_n\to 0$ almost surely by the Borel-Cantelli lemmas. Independence and $\mathbb{E}{Z}_n=1$ giveBut the independent events $\{Y_n=1\}$ have divergent probability sum, so the second Borel-Cantelli lemma makes them occur infinitely often. Thus these conditional expectations do not converge almost surely to zero.

The Skorokhod embedding theorem states that if $\mu$ is a centered probability law on $\mathbb{R}$ with finite second moment, there is a Brownian stopping time $T$ such that $B_T\sim \mu$, the stopped process $(B_{t\wedge T})$ is uniformly integrable, and $\mathbb{E}T=\int x^2\mu (dx)$.

Construct the times inductively. Suppose $(B_{T_0},\dots ,B_{T_n})$ has the law of $(S_0,\dots ,S_n)$. Conditional on the past, the martingale increment $S_{n+1}-S_n$ has mean zero and finite second moment. Apply the conditional form of the Skorokhod embedding theorem to this regular conditional law, using the fresh Brownian motion $B_{T_n+t}-B_{T_n}$ supplied by the Strong Markov property. This gives a stopping time increment ${\tau }_{n+1}$ and $T_{n+1}=T_n+{\tau }_{n+1}$ such that the next Brownian increment has the required conditional law. Induction provesfor every $k$.