Singular mean-field backward stochastic Volterra integral equations in infinite dimensional spaces

TL;DR

This paper establishes the well-posedness of singular mean-field backward stochastic Volterra integral equations in infinite-dimensional spaces, providing foundational lemmas and extending results to forward equations with applications in control and finance.

Contribution

It introduces new existence and uniqueness results for singular mean-field MF-BSVIEs and MF-FSVIEs in infinite dimensions, along with foundational lemmas and practical applications.

Findings

Proved existence and uniqueness of solutions for singular MF-BSVIEs.

Extended analysis to singular MF-FSVIEs with solvability results.

Applied theoretical results to derive stochastic maximum principles.

Abstract

This paper investigates the well-posedness of singular mean-field backward stochastic Volterra integral equations (MF-BSVIEs) in infinite-dimensional spaces. We consider the equation: \[X(t) = \Psi(t) + \int_t^b P\big(t, s, X(s), \aleph(t, s), \aleph(s, t), \mathbb{E}[X(s)], \mathbb{E}[\aleph(t, s)], \mathbb{E}[\aleph(s, t)]\big) ds - \int_t^b \aleph(t, s) dB_s, \] where the focus lies on establishing the existence and uniqueness of adapted M-solutions under appropriate conditions. A key contribution of this work is the development of essential lemmas that provide a rigorous foundation for analyzing the well-posedness of these equations. In addition, we extend our analysis to singular mean-field forward stochastic Volterra integral equations (MF-FSVIEs) in infinite-dimensional spaces, demonstrating their solvability and unique adapted solutions. Finally, we strengthen our theoretical results by applying them to derive stochastic maximum principles, showcasing the practical relevance of the proposed framework. These findings contribute to the growing body of research on mean-field stochastic equations and their applications in control theory and mathematical finance.