Some Results on Backward Stochastic Differential Equations of Fractional Order

TL;DR

This paper investigates the well-posedness of Caputo fractional backward stochastic differential equations of order between 0.5 and 1, introducing a new weighted norm to establish fundamental results and linking solutions to stochastic Volterra integral equations.

Contribution

It introduces a novel weighted norm in the analysis of Caputo fBSDEs and establishes the equivalence between solutions and stochastic Volterra integral equations.

Findings

Established well-posedness of Caputo fBSDEs with Lipschitz coefficients.

Introduced a new weighted norm for analyzing fractional stochastic equations.

Proved the equivalence between solutions of Caputo fBSDEs and stochastic Volterra integral equations.

Abstract

In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order $\alpha \in (\frac{1}{2},1)$ whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution.