Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 30 / 2

Fix the initial state $x$, or a prescribed initial distribution. A weak stochastic solution consists of a filtered probability space, a Brownian motion $W$ in that filtration, and a continuous adapted process $X$, with the prescribed initial law, such that the integrals exist andalmost surely for every $t$. The integrability conditions on each finite interval are ${\int }_0^t|b(X_s)|ds<\infty$ and ${\int }_0^t\sigma (X_s{)}^{2}ds<\infty$ almost surely. The probability space and driving Brownian motion are part of what may be chosen.

A strong stochastic solution is constructed on a space carrying a specified driving Brownian motion and specified initial variable. It satisfies the same equation and is adapted to the completed filtration generated by that initial variable and the Brownian motion. Equivalently, it is a nonanticipating measurable function of those data, requiring no additional randomness. For a random initial variable, it is independent of future Brownian increments, as required by the Brownian property of the enlarged filtration.

Uniqueness in law means that any two weak stochastic solutions with the same prescribed initial distribution have the same distribution as random elements of the continuous-path space. Their probability spaces and driving Brownian motions may differ.

Pathwise uniqueness means that if two solutions are defined on the same filtered probability space, driven by the same Brownian motion, and have the same initial variable almost surely, thenThis is indistinguishability of stochastic processes, a statement about coupled sample paths. It is stronger than requiring only equality of their distributions.

Let $B=\Vert b{\Vert }_{\infty }$ and define the scale function of a one-dimensional diffusionSince $b$ is continuous, $g\in C^2$, andThus $g$ is strictly increasing. The Itô formula for the additive-noise equation givesThe integrand is locally square-integrable: a continuous path $X$ has compact range on every finite interval, where $g^{\prime }$ is bounded. HenceThis is the scale transform for an additive-noise diffusion. Its range is the open interval $I=g(\mathbb{R})$, which need not be all of $\mathbb{R}$.

On $I=g(\mathbb{R})$ define $h(y)=g^{\prime }(g^{-1}(y))$. Differentiation givesThe mean value theorem proves Lipschitz continuity of $h$ on $I$, with constant $2B$.

If an endpoint of $I$ is finite, the Lipschitz bound gives a finite limiting value of $h$ there. That value must be zero. Otherwise $h$ would be bounded below by a positive number near the endpoint, andwould make $g^{-1}$ approach a finite limit, contradicting its tending to $+\infty$ or $-\infty$. The zero extension of a scale diffusion coefficient at finite endpoints therefore gives a globally Lipschitz function $\bar{h}$ on $\mathbb{R}$: set it to zero beyond each finite endpoint and retain $h$ on $I$.

Use the following standard global existence theorem for stochastic differential equations with Lipschitz coefficients: globally Lipschitz drift and diffusion coefficients give a nonexplosive, pathwise unique strong stochastic solution for every prescribed initial state and driving Brownian motion. Global Lipschitz continuity also gives the required linear-growth bound. Apply it toIt remains to check that $Y$ stays in $I$, so the inverse transform is defined.

Before the first boundary time, put $X=g^{-1}(Y)$. Since $(g^{-1}{)}^{\prime }=1/h$ and $(g^{-1}{)}^{\prime \prime }=-h^{\prime }/h^2$, the Itô formula, stopped inside compact subintervals of $I$, yieldsFor each finite $T$, up to that boundary time,A finite endpoint of $I$ would require $X$ to diverge, which this bound excludes. Therefore $Y$ stays in $I$ at every finite time, and $X=g^{-1}(Y)$ is a global strong stochastic solution.

Finally, any two solutions driven by the same $W$ transform into solutions of the globally Lipschitz $Y$ equation. Its pathwise uniqueness makes the transforms, and hence their inverses, indistinguishable. Thus