On matching diffusions, Laplace transforms and partial differential equations

TL;DR

This paper introduces a method of intertwining diffusions using Feynman-Kac operators to compute Laplace transforms and solve PDEs, providing new formulas for stochastic process distributions and hyperbolic PDE solutions.

Contribution

It develops a generalized intertwining technique for diffusions, enabling new analytical solutions for Laplace transforms and hyperbolic PDEs, with applications to stochastic process distributions.

Findings

Derived new formulas for Laplace transforms of stochastic processes.

Established a Feynman-Kac-based method for hyperbolic PDEs.

Applied the method to generalized Bessel processes and wave equations.

Abstract

We present the idea of intertwining of two diffusions by Feynman-Kac operators. We present some variations and implications of the method and give examples of its applications. Among others, it turns out to be a very useful tool for finding the expectations of some functionals of diffusions, especially for computing the Laplace transforms of stochastic processes. The examples give new results on marginal distributions of many stochastic processes including a generalized squared Bessel processes and joint distribution for squared Bessel bridge and its integral - the close formulae of the Laplace transforms are presented. We finally present a general version of the method and its applications to PDE of the second order. A new dependence between diffusions and solutions of hyperbolic partial differential equations is presented. In particular, the version of Feynman-Kac representation for hyperbolic PDE is given. It is presented, among others, the simple form of Laplace transform of wave equation with axial symmetry.