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

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 $k$.