A probabilistic point of view for the exact or approximated computation of the solution to Kolmogorov hypoelliptic equations

In this work, we propose a method for solving Kolmogorov hypoelliptic equations based on Fourier transform and Feynman-Kac formula. We first explain how the Feynman-Kac formula can be used to compute the fundamental solution to parabolic equations with linear or quadratic potential. Then applying these results after a Fourier transform we deduce the computation of the solution to a a first class of Kolmogorov hypoelliptic equations. Then we solve partial differential equations obtained via Feynman-Kac formula from the Ornstein-Uhlenbeck generator. Also, a new small time approximation of the solution to Kolmogorov hypoelliptic equations is provided. We finally present the results of numerical experiments to check the practical efficiency of this approximation.

Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators

In this paper we study time-inhomogeneo