Backward Stochastic Volterra Integral Equations (BSVIEs) have evolved into one of the most powerful and flexible mathematical frameworks for modeling systems with memory, time inconsistency, nonlinear dynamics, and path dependent uncertainty. Spanning foundational theory through cutting edge research, this comprehensive monograph offers the first unified and rigorous treatment of BSVIEs in their full generality.This landmark volume develops the analytic core of the subject—from classical stochastic calculus and Malliavin techniques to the modern theory of M-solutions, adapted solutions, comparison principles, and representation PDEs. Building systematically from BSDEs and forward Volterra equations, the book presents the most complete framework to date for well-posedness, stability, regularity, and qualitative analysis of BSVIEs, including equations with non-uniform, quadratic, and superquadratic generators.Beyond theory, the manuscript showcases the profound role of BSVIEs across contemporary applied mathematics. Readers will find deep connections to optimal control with memory, dynamic risk measures, recursive utilities, rough volatility models, mean-field interactions, stochastic games, and nonlinear pricing. The book also elaborates maximum principles, duality structures, and variational methods that place BSVIEs at the center of modern stochastic control and mathematical finance.Key features include:A complete and rigorous development of Type I, Type II, and anticipated BSVIEsDetailed well-posedness theory under Lipschitz, Osgood, quadratic, and superquadratic growthModern tools including Malliavin calculus, BMO martingales, nonlocal PDE representations, and comparison principlesFull treatment of mean-field BSVIEs and McKean–Vlasov interactionsOptimal control of systems with memory: adjoint equations, variational inequalities, and maximum principlesApplications to finance, recursive utilities, risk measures, equilibrium pricing, and rough volatilityOver 200 references connecting classical Volterra theory to the most recent advances (up to 2025)Comprehensive, rigorous, and forward-looking, this monograph is an essential reference for graduate students, researchers, and practitioners working in stochastic analysis, optimal control, mathematical finance, engineering, and applied probability. It not only consolidates the existing theory of BSVIEs but also lays the groundwork for their next decade of development.
