A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

Abstract

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite family of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semi-linear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of timeinconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.

On quadratic multidimensional type-I BSVIEs, infinite families of BSDEs and their applications

“…This paper studies type-I extended backward stochastic Volterra integral equations, BSVIEs for short, as recently revisited in Hernández and Possamaï [20]. Let X be the solution to a drift-less stochastic differential equation, SDE for short, under a probability measure P, F be the P-augmentation of the filtration generated by X, see Section 2.1 for details.…”

“…A prerequisite for rigorously introducing these processes is some regularity of the solution. Indeed, the regularity of s −→ (Y s , Z s ) in combination with the pathwise continuity of Y and the introduction of a derivative of s −→ Z s , as proposed in [20], are sufficient for the analysis. We also remark that, as we work with a general filtration F, the additional process N corresponds to a martingale process which is P-orthogonal to X.…”

“…In this work, we focus on the, to the best of knowledge, not studied case of multidimensional type-I BSVIEs with quadratic generators. Notably, we extend the analysis in [20] to the multidimensional quadratic case by exploiting the fact that the well-posedness of (1.1) is equivalent to that of a infinite family of backward stochastic differential equations, BSDEs for short.…”

“…We also present and discuss a type of flow property satisfied by this family of BSVIEs. As a preliminary step, we establish the well-posedness of a class of infinite families of BSDEs, as introduced in Hernández and Possamaï [20], which are of interest in their own right. Our approach relies on the strategy developed by Tevzadze [49] for quadratic BSDEs and the treatment of Lipschitz extended type-I BSVIEs in [20].…”

“…As a preliminary step, we establish the well-posedness of a class of infinite families of BSDEs, as introduced in Hernández and Possamaï [20], which are of interest in their own right. Our approach relies on the strategy developed by Tevzadze [49] for quadratic BSDEs and the treatment of Lipschitz extended type-I BSVIEs in [20]. We motivate their analysis of both of these objects by a series of practical applications.…”