Picard Approximation of a Singular Backward Stochastic Nonlinear Volterra Integral Equation - Ahmadova, Arzu; Mahmudov, Nazim I. (Springer Basel Ag, 2024)
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 is an element of ( 1 2 , 1 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (\frac{1}{2},1)$$\end{document} under a weaker condition than Lipschitz one in Hilbert space.
Keywords: Singular backward stochastic equations; Backward stochastic nonlinear Volterra integral equation; Existence and uniqueness; Picard iteration; Adapted process; Carath & eacute;odory conditions