Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 5

A weak solution consists of a filtered probability space carrying a Brownian motion $W$ and an adapted continuous process $X$ satisfyingThe probability space and Brownian motion are part of the unknown solution.

A strong solution is adapted to the augmented filtration generated by a prescribed Brownian motion and initial condition; equivalently, it is constructed measurably from that given noise.

Uniqueness in law means that any two weak solutions with the same initial distribution have the same probability distribution on path space.

Pathwise uniqueness means that two solutions on the same filtered space, driven by the same Brownian motion and with the same initial value, are indistinguishable.

Define the scale function of a one-dimensional diffusionThen $g^{\prime }>0$, so $g$ is strictly increasing, andBy Itô formula,so $Y=g(X)$ is a local martingale.

LetDifferentiating and using the scale equation givesSince $b$ is bounded, $h$ has global Lipschitz continuity. Thushas a pathwise unique strong solution by the standard Lipschitz existence-and-uniqueness theorem for a stochastic differential equation. Applying the deterministic inverse $g^{-1}$ gives a strong solution $X$, and uniqueness of $Y$ gives pathwise uniqueness of $X$.