Kolmogorov backward equation

Summary

The Kolmogorov backward equation is a parabolic partial differential equation satisfied by expectations $u(t, x) = E[f(X_t) \mid X_0 = x]$ of a diffusion process. Its operator is the infinitesimal generator of the diffusion, acting on the starting point $x$. It links stochastic differential equations with partial differential equations and is the basis of many probabilistic formulas for solutions of such equations.