Nonlinear Fractional Backward Doubly Stochastic Differential Equations with Hurst Parameter in (1/2,1)

TL;DR

This paper develops a new mathematical framework linking nonlinear backward doubly stochastic differential equations driven by both standard and fractional Brownian motions to classical equations, using advanced stochastic calculus and transformations.

Contribution

It introduces a novel Itô formula for mixed stochastic integrals and applies the Doss-Sussman transformation to connect complex equations involving fractional Brownian motion to standard stochastic differential equations.

Findings

Established a special Itô formula for mixed stochastic integrals.

Linked backward doubly stochastic differential equations with fractional Brownian motion to classical equations.

Extended the approach to nonlinear stochastic partial differential equations driven by fractional Brownian motions.

Abstract

We first state a special type of It\^o formula involving stochastic integrals of both standard and fractional Brownian motions. Then we use Doss-Sussman transformation to establish the link between backward doubly stochastic differential equations, driven by both standard and fractional Brownian motions, and backward stochastic differential equations, driven only by standard Brownian motions. Following the same technique, we further study associated nonlinear stochastic partial differential equations driven by fractional Brownian motions and partial differential equations with stochastic coefficients.