Importance sampling for Kolmogorov backward equations

Abstract

The solution of the Kolmogorov backward equation is expressed as a functional integral by means of the Feynman–Kac formula. The expectation value is approximated as a mean over trajectories. In order to reduce the variance of the estimate, importance sampling is utilized. From the optimal importance density, a modified drift function is derived which is used to simulate optimal trajectories from an Ito equation. The method is applied to option pricing and the simulation of transition densities and likelihoods for diffusion processes. The results are compared to known exact solutions and results obtained by numerical integration of the path integral using Euler transition kernels. The importance sampling leads to strong variance reduction, even if the unknown solution appearing in the drift is replaced by known reference solutions. In models with low-dimensional state space, the numerical integration method is more efficient, but in higher dimensions it soon becomes infeasible, whereas the Monte Carlo method still works.