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Infinite Horizon Optimal Control of Forward-Backward Stochastic Volterra Equations with Delay

Ibtissem Djaber, Hafiane Nawel, Samia Yakhlef

Abstract

We consider an optimal control problem for infinite horizon systems governed by coupled forward-backward stochastic Volterra integral equations with delay. Using Hida-Malliavin calculus, we prove both sufficient and necessary maximum principles for optimal control of such systems. We establish existence and uniqueness results for a class of infinite horizon backward stochastic Volterra integral equations (BSVIEs).

Infinite Horizon Optimal Control of Forward-Backward Stochastic Volterra Equations with Delay

Abstract

We consider an optimal control problem for infinite horizon systems governed by coupled forward-backward stochastic Volterra integral equations with delay. Using Hida-Malliavin calculus, we prove both sufficient and necessary maximum principles for optimal control of such systems. We establish existence and uniqueness results for a class of infinite horizon backward stochastic Volterra integral equations (BSVIEs).

Paper Structure

This paper contains 16 sections, 6 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Setting of the Problem
  3. Forward Equation
  4. Backward Equation
  5. Assumptions H1
  6. Hamiltonian and Adjoint Equations
  7. Function Spaces
  8. Elements of Malliavin Calculus
  9. Sufficient Maximum Principle
  10. Necessary Maximum Principle
  11. Linearized Equations
  12. First Variation of the Cost Functional
  13. Existence and uniqueness for infinite-horizon BSVIEs
  14. Problem formulation
  15. Functional framework
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $\hat{u} \in \mathcal{U}$ with corresponding solutions $\hat{X}(t)$, $(\hat{Y}(t), \hat{Z}(t,\cdot), \hat{K}(t,\cdot,\cdot))$, $(\hat{p}(t), \hat{q}(t,\cdot), \hat{r}(t,\cdot,\cdot))$ and $\hat{\lambda}(t)$ of equations (b1), (b2), (eq3.2) and (p) respectively. Suppose that: Then, $\hat{u}$ is an optimal control for our problem. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (12)

  • Remark 2.1
  • Theorem 3.1: Sufficient Maximum Principle
  • proof
  • Remark 3.2
  • Lemma 4.1: Linearized forward equation
  • Lemma 4.2: Linearized backward equation
  • Lemma 4.3: First variation formula
  • proof
  • Theorem 4.4: Necessary Maximum Principle
  • proof
  • ...and 2 more