A complex Feynman-Kac formula via linear backward stochastic differential equations

TL;DR

This paper introduces a complex backward stochastic differential equation framework to interpret linear first order complex PDEs, extending the real Feynman-Kac formula into the complex domain with unique solutions.

Contribution

It develops a novel complex BSDE approach to provide a probabilistic representation for linear first order complex PDEs, establishing existence, uniqueness, and analyticity of solutions.

Findings

Existence and uniqueness of regular solutions to complex BSDEs.

Representation of complex PDE solutions via complex FBSDEs.

Extension of the Feynman-Kac formula to complex PDEs.

Abstract

A complex notion of backward stochastic differential equation (BSDE) is proposed in this paper to give a probabilistic interpretation for linear first order complex partial differential equation (PDE). By the uniqueness and existence of regular solutions to complex BSDE, we deduce that there exists a unique classical solution $\{\mathbb{U}(t,x)$ to complex PDE and $\{\mathbb{U}(t,x)$ is analytic in $x$ for each $t$. Thus we extend the well known real Feynman-Kac formula to a complex version. It is stressed that our complex BSDE corresponds to a linear PDE without the second order term.