Stochastic Functional Differential Equations and Feynman-Kac Formula

TL;DR

This paper develops representation formulas for degenerate stochastic functional differential equations, extending Feynman-Kac type results to applications like bacterial motility and path-dependent financial models.

Contribution

It introduces new Feynman-Kac formulae for degenerate SFDEs with boundary conditions, expanding the theoretical framework and practical applications.

Findings

Derived formulas for first exit time distributions.

Applied results to bacterial motility models.

Extended Feynman-Kac formulae to degenerate SFDEs.

Abstract

In the framework of stochastic functional differential equations (SFDE's) and the corresponding calculus developed in the recent years by F. Yan and S. Mohammed, we provide a series of representation formulae for a variety of highly degenerate functional differential equations of the type of the Feynman-Kac formulae. More precisely, we study the stochastic process satisfying regular SFDE's with killing and absorbing boundary, we give the differential equation to be solved in order to compute the distribution of the first exit time from a regular domain and apply our results to a model describing bacterial motility and to the derivation of a path-dependent Black-Scholes equation.