Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 38 / 3

The existence and pathwise uniqueness theorem for a stochastic differential equation states that locally Lipschitz drift $b$ and diffusion $\sigma$ on an open domain give a pathwise unique strong solution of a stochastic differential equation up to a maximal lifetime $\xi$. The initial value is measurable with respect to the initial sigma-algebra, and the filtration satisfies the usual conditions for a filtration. On ${\mathbb{R}}^d$, if $T_n$ are exits from increasing balls, then $T_n\uparrow \xi$; a finite lifetime entails escape from every compact subset. The coefficients need not satisfy a global growth bound for this local result.

One construction truncates the coefficients outside each compact region to globally Lipschitz coefficients, solves the resulting global equations, and patches their solutions until the exits. Pathwise uniqueness makes solutions from larger truncations agree on the smaller stopped interval. Their union is the maximal local solution of a stochastic differential equation, and any extension would contradict the exhaustion of the domain by the exit times.

Use the Itô formula up to $T_n$:For fixed $n$, the stopped stochastic integrand is bounded on each finite horizon, so its integral has expectation zero. The linear growth condition for an SDE and $2xb(x)\le x^2+b(x{)}^2$ giveWriting $u_n(t)=\mathbb{E}{X}_{t\wedge T_n}^2$, and bounding the active-time integrand by the stopped square, yieldsThe Gronwall inequality gives a bound independent of $n$:On $\{T_n\le t\}$, continuity gives $|X_{t\wedge T_n}|=n$. The Markov inequality therefore givesSince $T_n\uparrow \xi$, the event $\{\xi \le t\}$ is the intersection of these decreasing events and has probability zero. Taking integer $t$ proves $\xi =\infty$ almost surely. This proves global existence with locally, rather than globally, Lipschitz coefficients under the stated growth bound.