Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations

TL;DR

This paper develops Feynman formulae for certain Feller semigroups, providing a practical method for approximating functional integrals and simulating Feller processes through iterated integrals.

Contribution

It introduces new Feynman formulae for Feller semigroups, enabling effective approximation and simulation of associated stochastic processes.

Findings

Feynman formulae derived for specific Feller semigroups

Finite-dimensional integrals approximate functional integrals in Feynman--Kac formulae

Method facilitates simulation of Feller processes

Abstract

A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of $n$-fold iterated integrals of some elementary functions as $n\to\infty$. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman--Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae give an effective tool to calculate functional integrals with respect to probability measures generated by these Feller processes and, in particular, to obtain simulations of Feller processes.