Abstract

Stochastic differential equations such as $dX_t=b(t,X_t)dt+dW_t$ with irregular drift $b$ arise naturally in models with highly singular interaction, but in this setting the meaning of the solution becomes unclear. In particular, when the drift is only a distribution in space, even the expression $\int_0^t  b(s,X_s)ds$ is not classically well-defined.

The goal of this talk is to explain how one can nevertheless give a rigorous meaning to such an equation and how it may be approximated by classically defined objects. We will start with the definition of the solution and how one needs to solve a backwards PDE to give meaning to it before moving on to numerical approximation results.