A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

TL;DR

This paper develops a nonlinear Kolmogorov equation for stochastic delay differential equations with jumps, establishing a Feynman-Kac representation in an infinite-dimensional setting with applications to systems with memory.

Contribution

It introduces a novel nonlinear Kolmogorov equation framework for stochastic delay equations with jumps and derives a Feynman-Kac formula in an infinite-dimensional space.

Findings

Solution is an $L^2$-valued Markov process

Uniqueness proven under standard Lipschitz and linear growth conditions

Derived a nonlinear Feynman-Kac representation theorem

Abstract

We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an $L^2-$valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non--linear Feynman--Kac representation theorem under mild assumptions of differentiability.