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Explicit Solution of Infinite-Horizon Linear Backward Stochastic Volterra Integral Equations

Samia Yakhlef, Hilel Ardjan

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.

Explicit Solution of Infinite-Horizon Linear Backward Stochastic Volterra Integral Equations

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.
Paper Structure (29 sections, 9 theorems, 49 equations)

This paper contains 29 sections, 9 theorems, 49 equations.

Table of Contents

  1. Keywords:
  2. MSC(2020):
  3. Introduction
  4. Mathematical Framework and Assumptions
  5. Probability Space
  6. Weighted Function Spaces
  7. Adapted M-Solution
  8. Assumptions
  9. Main Results
  10. Measure Transformation
  11. Girsanov Transformation
  12. Transformed Equation
  13. Infinite Horizon Resolvent Kernel Theory
  14. Definition and Construction
  15. Resolvent Equation
  16. ...and 14 more sections

Key Result

Lemma 3.1

Under $\mathbb{Q}$:

Theorems & Definitions (13)

  • Definition 2.1: Weighted spaces for infinite horizon
  • Definition 2.2: Adapted M-Solution
  • Lemma 3.1: Girsanov theorem for infinite horizon
  • Lemma 3.2: Convergence of the infinite series
  • Lemma 3.3: Resolvent equation
  • Theorem 3.4: Existence and uniqueness
  • Theorem 3.5: Explicit solution of infinite-horizon BSVIE
  • Remark 3.6
  • Corollary 3.7: Deterministic coefficients
  • Definition A.1: Malliavin-Sobolev spaces on infinite horizon
  • ...and 3 more