BSDEs generated by fractional space-time noise and related SPDEs

TL;DR

This paper studies backward stochastic differential equations driven by fractional space-time noise, establishing existence, uniqueness, and explicit solutions, and connecting them to linear SPDEs with colored noise.

Contribution

It introduces a new class of BSDEs driven by weighted fractional Brownian fields and provides conditions for their well-posedness and explicit solution formulas.

Findings

Existence and uniqueness of solutions under specific Hurst parameter conditions

Explicit formulas for solution components Y and Z

Probabilistic representations for certain linear SPDEs

Abstract

This paper is concerned with the backward stochastic differential equations whose generator is a weighted fractional Brownian field: $Y_t=\xi+\int_t^T Y_s W (ds,B_s) -\int_t^T Z_sdB_s$, $0\le t\le T$, 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.