Picard approximation of a singular backward stochastic nonlinear Volterra integral equation

TL;DR

This paper proves exponential convergence of Picard iterations for BSDEs with Lipschitz nonlinearities and establishes existence and uniqueness of solutions for a class of singular backward stochastic Volterra integral equations of order between 0.5 and 1 under weaker conditions.

Contribution

It introduces a fundamental lemma that ensures global existence and uniqueness of solutions to singular BSVIEs under less restrictive conditions than Lipschitz continuity.

Findings

Picard iterations converge exponentially fast for BSDEs with Lipschitz nonlinearities.

Established global existence and uniqueness for singular BSVIEs of order in (0.5,1).

Proved a fundamental lemma under weaker conditions than Lipschitz.

Abstract

Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. In this paper we prove that Picard iterations of BSDEs with globally Lipschitz continuous nonlinearities converge exponentially fast to the solution. Our main result in this paper is to establish a fundamental lemma to prove the global existence and uniqueness of an adapted solution to a singular backward stochastic nonlinear Volterra integral equation (for short singular BSVIE) of order $\alpha \in (\frac{1}{2},1)$ under a weaker condition than Lipschitz one in a Hilbert space.