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Stochastic maximum principle for time-changed forward-backward stochastic control problem with Lévy noise

Jingwei Chen, Jun Ye, Feng Chen

Abstract

This paper establishes a stochastic maximum principle for optimal control problems governed by time-changed forward-backward stochastic differential equations with Lévy noise. The system incorporates a random, non-decreasing operational time (the inverse of an $α$-stable subordinator) to model phenomena like trapping events and subdiffusion. Using a duality transformation and the convex variational method, we derive necessary and sufficient conditions for optimality, expressed through a novel set of adjoint equations. Finally, the theoretical results are applied to solve an explicit cash management problem under stochastic recursive utility.

Stochastic maximum principle for time-changed forward-backward stochastic control problem with Lévy noise

Abstract

This paper establishes a stochastic maximum principle for optimal control problems governed by time-changed forward-backward stochastic differential equations with Lévy noise. The system incorporates a random, non-decreasing operational time (the inverse of an -stable subordinator) to model phenomena like trapping events and subdiffusion. Using a duality transformation and the convex variational method, we derive necessary and sufficient conditions for optimality, expressed through a novel set of adjoint equations. Finally, the theoretical results are applied to solve an explicit cash management problem under stochastic recursive utility.

Paper Structure

This paper contains 8 sections, 10 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Preliminaries
  4. Duality
  5. Stochastic Maximum Principle
  6. Convex variational method
  7. Main results
  8. Application to a cash management problem

Key Result

Lemma 2.1

(Itô formula for time-changed Lévy noise) Let $D_{t}$ be an RCLL subordinator and $E_{t}$ its inverse process. Define a filtration $\{\mathcal{G}_{t}\}_{t\geq0}$ by $\mathcal{G}_{t}=\mathcal{F}_{E_{t}}$. Let $X$ be a process defined as follows: where $f,k,g,h$ are measurable functions such that all integrals are defined. Here $c$ is the maximum allowable jump size. Then, for all $F:\mathbb{R}_{+}

Theorems & Definitions (16)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 6 more