Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 2

Write ${\Delta }_{i}M=M_{t_i}-M_{t_{i-1}}$ and $Q={\sum }_i({\Delta }_{i}M{)}^2$. The identityshows, using the martingale-difference orthogonality, thatAlso $\mathbb{E}[Q]=\mathbb{E}[M_t^2]-\mathbb{E}[M_0^2]\le C^2$. Hence $\mathbb{E}[Q^2]\le 10C^4$, which in particular proves the requested bound

Because $\Vert b^{\prime }{\Vert }_{\infty }<\infty$, the function $b$ is Lipschitz continuous. the Picard-Lindelof theorem, proved by iteration onconverges uniformly on every compact interval. The usual factorial estimate proves convergence for arbitrary interval length, and the Gronwall inequality proves uniqueness. Thus there is a unique global continuous solution.

For $w=W$, induction shows that $X_t^{(n)}$ is ${\mathcal{F}}_t$-measurable: its value uses only $(W_s{)}_{s\le t}$ and earlier iterates up to time $t$. The pointwise limit $X_t$ is therefore ${\mathcal{F}}_t$-measurable. Hence $X$ is adapted to $({\mathcal{F}}_t)$.

Let ${\tau }_n$ be the first exit from a compact interval on which $H<n$, chosen so that ${\tau }_n\uparrow T$, and let $m=\inf H>-\infty$. By Itô formula,The assumption gives the drift bound $\frac{1}{2}{H}^{\prime \prime }-(H^{\prime }{)}^2\le C$. After stopping,If $h_n=\inf \{H(y):y\text{ lies outside the }n\text{th compact interval}\}$, then $h_n\to \infty$ because $H$ is coercive, andThus $\mathbb{P}(T\le t)=0$ for every $t$, and $T=\infty$ almost surely.