Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

Abstract

In this paper, we study extended backward stochastic Volterra integral equations (EBSVIEs, for short). We establish the well-posedness under weaker assumptions than the literature, and prove a new kind of regularity property for the solutions. As an application, we investigate, in the open-loop framework, a time-inconsistent stochastic recursive control problem where the cost functional is defined by the solution to a backward stochastic Volterra integral equation (BSVIE, for short). We show that the corresponding adjoint equations become EBSVIEs, and provide a necessary and sufficient condition for an open-loop equilibrium control via variational methods.

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

“…For this, the definition of the space (H , • H ), notably the space H 2,2 (R n×d ), and (v) Let us remark that Assumption 4.1. (i), being an assumption on the data of the BSVIE, is easier to verify in practice compare to the regularity required in [27]. Certainly, our results would still hold true if we require the differentiability of data (ξ, f ) with respect to the parameter s in the L 2 (resp.…”

“…Likewise, Hamaguchi [26,27] studied a time-inconsistent control problem where the cost functional is defined by the Y component of the solution of a type-I BSVIE (1.3), in which g depends on a control. Via Pontryagin's optimal principle, the author noticed that the adjoint equations correspond to an extended type-I BSVIE, as first introduced in Wang [53] in the context of generalising the celebrated Feynman-Kac formula.…”

“…This is a consequence of the fact that we work with a general filtration F. To the best of our knowledge, a theory for type-I BSVIEs, as general as the ones introduced above, remains absent in the literature. Moreover, such class of type-I BSVIEs has only been mentioned in [27,Remark 3.8] as an interesting generalisation of (1.5).…”

“…We highlight that Theorem 4.4 below guarantees a unique solution to (4.1) in the space H , for which the mapping s −→ Z s is assumed to be absolutely continuous, cf [27,. Section 2.1].…”

“…This allows us to define the diagonal of Z following [27, Section 2.1], and consequently, introduce the space H 2,2 (R n×d ). (iv) Moreover, not only does our approach identify the dynamics of (∂Y, ∂Z, ∂N ), but also, in the terminology of[27, Section 2.1], it does guarantee (Z t t ) t∈[0,T ] is the unique process that satisfies the (D)-property with respect to Z.EJP 26 (2021), paper 89.…”