BSDEs generated by fractional space-time noise and related SPDEs

Abstract

This paper is concerned with the backward stochastic differential equations whose generator is a weighted fractional Brownian field: $Y_t=\xi +{\int }_t^T{Y}_{s}W(ds,B_s)-{\int }_t^T{Z}_{s}d{B}_s$, $0\le t\le T$, where $W$ is a $(d+1)$-parameter weighted fractional Brownian field of Hurst parameter $H=(H_0,H_1,\cdots ,H_d)$, which provide probabilistic interpretations (Feynman-Kac formulas) for certain linear stochastic partial differential equations with colored space-time noise. Conditions on the Hurst parameter $H$ and on the decay rate of the weight are given to ensure the existence and uniqueness of the solution pair. Moreover, the explicit expression for both components $Y$ and $Z$ of the solution pair are given.