Semilinear backward doubly stochastic differential equations and SPDEs driven by fractional Brownian motion with Hurst parameter in $(0,1/2)$

Abstract

We study the existence of a unique solution to semilinear fractional backward doubly stochastic differential equation driven by a Brownian motion and a fractional Brownian motion with Hurst parameter less than 1/2. Here the stochastic integral with respect to the fractional Brownian motion is the extended divergence operator and the one with respect to Brownian motion is Itôʼs backward integral. For this we use the technique developed by R. Buckdahn (1994) [3] to analyze stochastic differential equations on the Wiener space, which is based on the Girsanov theorem and the Malliavin calculus, and we reduce the backward doubly stochastic differential equation to a backward stochastic differential equation driven by the Brownian motion. We also prove that the solution of semilinear fractional backward doubly stochastic differential equation defines the unique stochastic viscosity solution of a semilinear stochastic partial differential equation driven by a fractional Brownian motion.