Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 202 / 3 / b

Fix $X_0=x$ and a finite horizon $T$. Begin under Wiener measure $\mathbb{P}$ with $X_t=x+W_t$, and let $K$ bound $|b|$. The bounded progressively measurable process $b(X)$ is an admissible Itô integral integrand. PutThe Novikov condition follows from ${\int }_0^{T}b(X_s{)}^{2}ds\le K^{2}T$. With $d\mathbb{Q}=Z_{T}d\mathbb{P}$, the Girsanov theorem says that $B=W-{\int }_0^{\cdot }b(X_s)ds$ is Brownian motion. Therefore a weak stochastic solution exists, sinceConsistent finite-horizon densities on canonical continuous path space give a solution for all finite times. No uniform integrability at infinite time is needed for this extension.

For uniqueness in law, take any weak stochastic solution under a measure $\mathbb{Q}$ and reverse the change of measure:Again the Novikov condition holds. Under $\mathbb{P}$, $W=B+\int b(X_s)ds=X-x$ is Brownian motion, so the law of $X$ is the fixed shifted Wiener measure. Substitution of $dB=dX-b(X)dt$ givesUnder shifted Wiener measure, the stochastic integral on the right has a fixed measurable version as a functional of the canonical path. It can be defined by simple predictable processes converging in the integrand $L^2$ norm and an almost surely convergent subsequence of their integrals. Equivalence of the two finite-horizon measures preserves that version. Thus for every bounded measurable path functional $F$,The right-hand side depends only on $b,x,T$, not on the chosen solution. This proves uniqueness in law on every finite horizon, and hence on the whole continuous path space. This is Weak existence and uniqueness in law for an additive-noise SDE with bounded drift. For merely measurable $b$, one should not assume that naive deterministic left-endpoint samples of $b(X)$ converge; the predictable $L^2$ approximation is the appropriate definition.