Backward problems for stochastic differential equations on the Sierpinski gasket

TL;DR

This paper investigates backward stochastic differential equations on the Sierpinski gasket, establishing existence, uniqueness, and a Feynman-Kac formula for related semi-linear PDEs in a fractal setting.

Contribution

It introduces the first rigorous analysis of backward SDEs on the Sierpinski gasket, including exponential integrability and a Feynman-Kac representation.

Findings

Proved existence and uniqueness of backward SDE solutions on the gasket.

Established exponential integrability of quadratic processes.

Derived a Feynman-Kac formula for semi-linear PDEs on fractals.

Abstract

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section [sec:-1]) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman-Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established.