[Paper Review] Explicit Solution of Infinite-Horizon Linear Backward Stochastic Volterra Integral Equations

Samia Yakhlef, Hilel Ardjan|arXiv (Cornell University)|Mar 16, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper extends explicit solution theory for linear backward stochastic Volterra integral equations to infinite horizon, constructing an infinite-horizon resolvent kernel and providing Malliavin-based representations for solution components via a Girsanov transform.

ABSTRACT

We study linear backward stochastic Volterra integral equations (BSVIEs) on the infinite time horizon. By introducing weighted function spaces with exponential decay, we establish existence and uniqueness of adapted M-solutions. We construct an infinite-horizon resolvent kernel and derive explicit formulas for the solution components (Y,Z,K) using a Girsanov transformation and Hida-Malliavin calculus. The results extend the finite-horizon theory of Hu and Oksendal to the infinite horizon framework.

Motivation & Objective

• Motivate the study of backward stochastic Volterra integral equations with memory on an infinite horizon.
• Establish existence and uniqueness of adapted M-solutions in weighted, exponentially decaying spaces.
• Derive explicit solution representations for (Y,Z,K) using an infinite-horizon resolvent kernel and measure transformation.
• Extend finite-horizon results of Hu and Øksendal to the infinite-horizon setting.
• Provide Malliavin calculus-based representations for the Z and K components under the transformed measure.

Proposed method

• Introduce weighted function spaces with exponential decay to handle infinite horizon integrability.
• Construct the infinite-horizon resolvent kernel Psi(t,s) as a series Psi = sum Phi^(n) and prove its convergence.
• Apply a Girsanov transformation to remove Z and K from the drift, defining an equivalent measure Q.
• Obtain an explicit Y(t) in terms of conditional expectations under Q and the resolvent kernel (Equation (5)).
• Use Clark–Ocone type Malliavin representations to express Z(t,s) and K(t,s,zeta) under Q (Equations (7)-(8)).
• Provide deterministic simplifications in the deterministic-coefficient case (Corollary 3.7).

Experimental results

Research questions

• RQ1 How can linear BSVIEs be solved on an infinite time horizon with memory effects?
• RQ2 Can an infinite-horizon resolvent kernel be constructed and used to obtain explicit solution formulas?
• RQ3 How can Malliavin calculus and a Girsanov transformation yield explicit representations for Z and K in this setting?
• RQ4 What are the conditions ensuring existence, uniqueness, and convergence in weighted, exponentially decaying spaces?
• RQ5 How do results connect to the finite-horizon theory of Hu and Øksendal and extend it to infinite horizon?

Key findings

• An adapted M-solution exists uniquely in the weighted spaces with exponential decay.
• An infinite-horizon resolvent kernel Psi(t,s) is constructed as a uniformly convergent series of iterated kernels Phi^(n).
• Y(t) admits an explicit representation under the transformed measure Q (Equation (5)).
• Z(t,s) and K(t,s,ζ) have explicit Malliavin representations under Q (Equations (7)-(8)).
• Deterministic-coefficient simplifications yield simpler Z and K expressions (Corollary 3.7).
• The results extend the finite-horizon theory of Hu and Øksendal to infinite horizon.