Fractional backward stochastic differential euqations and fractional backward variational inequalities

TL;DR

This paper investigates the existence and uniqueness of solutions for fractional backward stochastic differential equations driven by fractional Brownian motion, extending classical results to the fractional and multivalued case.

Contribution

It develops a rigorous approach for fractional backward stochastic differential equations driven by fractional Brownian motion and studies solutions for multivalued variants with subdifferential operators.

Findings

Established existence and uniqueness of solutions for fractional BDSDEs.

Extended classical BDSDE theory to fractional Brownian motion with Hurst > 1/2.

Analyzed solutions for multivalued backward stochastic differential equations.

Abstract

In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as: [{[c]{l}% -dY(t)= f(t,\eta(t),Y(t),Z(t))dt-Z(t)\delta B^{H}(t), \quad t\in[0,T], Y(T)=\xi,.] where $\eta$ is a stochastic process given by $\eta(t)=\eta(0) +\int_{0}^{t}\sigma(s) \delta B^{H}(s)$, $t\in[0,T]$, and $B^{H}$ is a fractional Brownian motion with Hurst parameter greater than 1/2. The stochastic integral used in above equation is the divergence-type integral. Based on Hu and Peng's paper, \textit{BDSEs driven by fBm}, SIAM J Control Optim. (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equation [{[c]{l} -dY(t)+\partial\varphi(Y(t))dt\ni f(t,\eta(t),Y(t),Z(t))dt-Z(t)\delta B^{H}(t),\quad t\in[0,T], Y(T)=\xi,.] where $\partial\varphi$ is a multivalued operator of subdifferential type associated with the convex function $\varphi$.