[Paper Review] Infinite horizon backward stochastic Volterra integral equations and discounted control problems

Yushi Hamaguchi|arXiv (Cornell University)|May 6, 2021

Stochastic processes and financial applicationsEconomics, Econometrics and Finance49 references15 citations

TL;DR

This paper establishes the existence and uniqueness of adapted M-solutions for infinite-horizon backward stochastic Volterra integral equations (BSVIEs) in a weighted L²-space, extending finite-horizon duality and variation-of-constant results. It applies these to infinite-horizon stochastic control problems with discounted cost functionals, deriving necessary and sufficient optimality conditions via Pontryagin's maximum principle with an adjoint equation formulated as an infinite-horizon BSVIE.

ABSTRACT

Infinite horizon backward stochastic Volterra integral equations (BSVIEs for short) are investigated. We prove the existence and uniqueness of the adapted M-solution in a weighted $L^2$-space. Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for a class of infinite horizon linear BSVIEs and prove a duality principle between a linear (forward) stochastic Volterra integral equation (SVIE for short) and an infinite horizon linear BSVIE in a weighted $L^2$-space. As an application, we investigate infinite horizon stochastic control problems for SVIEs with discounted cost functional. We establish both necessary and sufficient conditions for optimality by means of Pontryagin's maximum principle, where the adjoint equation is described as an infinite horizon BSVIE. These results are applied to discounted control problems for fractional stochastic differential equations and stochastic integro-differential equations.

Motivation & Objective

• To extend the theory of backward stochastic Volterra integral equations (BSVIEs) from finite to infinite time horizons.
• To prove existence and uniqueness of adapted M-solutions for infinite-horizon Type-I and Type-II BSVIEs in a weighted L²-space.
• To generalize duality principles and variation-of-constant formulas to the infinite-horizon setting with unbounded coefficients.
• To apply the infinite-horizon BSVIE framework to stochastic control problems with discounted cost functionals.
• To derive necessary and sufficient optimality conditions for infinite-horizon stochastic control of forward SVIEs using Pontryagin’s maximum principle with BSVIE adjoint equations.

Proposed method

• Introduces a discounted infinite-horizon BSVIE of Type-II form with a free term and driver depending on Z(s,t), incorporating a discount rate λ.
• Defines the adapted M-solution concept via the martingale representation theorem for Z(t,s) on 0 ≤ s ≤ t ≤ ∞.
• Proves existence and uniqueness of the adapted M-solution in a weighted L²-space by analyzing the trade-off between the weight η and discount rate λ.
• Establishes a duality principle between linear forward SVIEs and infinite-horizon linear Type-II BSVIEs in a weighted L²-space.
• Derives a variation-of-constant formula for infinite-horizon linear Type-I BSVIEs, generalizing finite-horizon results to unbounded coefficients.
• Applies the framework to infinite-horizon stochastic control problems by formulating the adjoint equation as an infinite-horizon BSVIE and deriving optimality conditions via Pontryagin’s principle.

Experimental results

Research questions

• RQ1 Under what conditions on the discount rate λ and weight η does an infinite-horizon BSVIE admit a unique adapted M-solution in a weighted L²-space?
• RQ2 How can the duality principle between forward SVIEs and infinite-horizon BSVIEs be extended from finite to infinite horizons with unbounded coefficients?
• RQ3 What is the infinite-horizon variation-of-constant formula for linear Type-I BSVIEs, and how does it generalize finite-horizon results?
• RQ4 How can Pontryagin’s maximum principle be applied to infinite-horizon stochastic control problems with discounted cost functionals when the state process is a forward SVIE?
• RQ5 What are the necessary and sufficient optimality conditions for infinite-horizon stochastic control of SVIEs with singular coefficients, and how are they expressed via BSVIE adjoint equations?

Key findings

• The paper proves the existence and uniqueness of the adapted M-solution for infinite-horizon BSVIEs in a weighted L²-space, with the well-posedness condition depending on the interplay between the discount rate λ and the weight η of the solution space.
• It establishes a variation-of-constant formula for infinite-horizon linear Type-I BSVIEs, extending results from Hu and Øksendal (2010) and Wang, Yong, and Zhang (2016) to unbounded coefficients.
• A duality principle is proven between linear forward SVIEs and infinite-horizon linear Type-II BSVIEs in a weighted L²-space, generalizing Yong (2006) to the infinite-horizon case.
• For infinite-horizon stochastic control problems with discounted cost functionals, the paper derives both necessary and sufficient optimality conditions via Pontryagin’s maximum principle, with the adjoint equation formulated as an infinite-horizon BSVIE.
• The optimality condition is expressed as a variational inequality involving the adjoint process and control, with the adjoint process satisfying an infinite-horizon anticipated BSDE of Itô–Volterra type.
• The framework is applied to fractional SDEs and stochastic integro-differential equations, showing that the infinite-horizon BSVIE approach can handle models with memory and singular coefficients.