Weak solution of a stochastic differential equation

A weak solution may choose its probability space and driving Brownian motion as part of the solution. It is weaker than a strong solution, which must be adapted to a prescribed Brownian motion.

There need not be a weak solution of a stochastic differential equation $d{X}_t=\sqrt{2\theta (X_t)}d{B}_t$, $X_0=0$, even when $\theta$ is a bounded Borel measurable function strictly positive everywhere. For the displayed example, any solution would be a continuous local martingale with $[X{]}_t=2{\int }_0^t\theta (X_s)ds\le 2t$, hence a square-integrable martingale. The occupation-times formula applied to $1_{\{0\}}$ gives $0={\int }_0^t{1}_{\{X_s=0\}}d[X{]}_s=2{\int }_0^t{1}_{\{X_s=0\}}ds$. Since $\theta (x)\le x^2$ away from zero, $\mathbb{E}{X}_t^2\le 2{\int }_0^t\mathbb{E}{X}_s^{2}ds$. The Gronwall inequality makes $X\equiv 0$, a contradiction. Thus a diffusion amplitude positive at a single problematic point cannot repair arbitrarily slow neighbouring motion. A local positive lower bound, as in the Brownian time change for a variable diffusion speed, rules out this example.

If $b:\mathbb{R}\to \mathbb{R}$ is bounded and measurable, then on every finite time intervalhas a weak solution and uniqueness in law. Starting with Wiener measure, the Novikov condition and Girsanov theorem add the drift. Applying the inverse change of measure to any weak solution recovers Wiener measure and identifies its law by the same pathwise density.