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 time-inconsistent 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.
Highlights
This paper is concerned with introducing a unified method to address the wellposedness of backward stochastic Volterra integral equations, BSVIEs for short
BSDEs of linear type were first introduced by Bismut [10, 11] as an adjoint equation in the Pontryagin stochastic maximum principle
The contemporary work of Davis and Varaiya [20]1 studied a precursor of a linear BSDE for characterising the value function and the optimal controls of stochastic control problems with drift control only
Summary
A natural extension of (1.1) arises by considering a collection of GT -measurable random variables (ξ(t))t∈[0,T ], referred in the literature of BSVIEs as the free term, as well as a generator g In such a setting, a solution to a BSVIE is a pair (Y·, Z··) of processes such that. In this paper we want to build upon the strategy devised in [29] and address the well-posedness of a general and novel class of type-I BSVIEs. We let X be the solution to a drift-less stochastic differential equation (SDE, for short) under a probability measure P, and F be the P-augmentation of the filtration generated by X, see Section 2.1 for details, and consider a tuple (Y··, Z··, N··), of appropriately F-adapted processes, which for any s ∈ [0, T ] satisfy, P−a.s. for any t ∈ [0, T ], the equation.
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