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Maximum principle for optimal control of infinite horizon stochastic difference equations driven by fractional noises

Yuecai Han, Yuhang Li

TL;DR

This paper tackles control of infinite-horizon discrete-time systems driven by fractional noises, formulating the problem with state dynamics $X_{n+1}=X_n+b(n,X_n,u_n)+\sigma(n,X_n,u_n)\xi_n^H$ and a cost represented by $J(u)=Y_0$ where $Y$ solves a BS$Delta$E discounted by $e^{-\lambda n^\gamma}$. It develops an infinite-horizon stochastic maximum principle by proving well-posedness of forward S$Delta$Es and backward BS$Delta$Es under tailored weighted norms $\|\cdot\|_{\lambda,\gamma}$ and establishing adjoint equations, a Hamiltonian, and a verification theorem. The main contributions include a rigorous SMP for fractional-noise-driven systems on infinite horizon, plus an explicit optimal-investment application that yields a computable control law under memory effects from fractional noise. These results extend stochastic control with fractional noise to infinite horizons, enabling robust optimization in long-horizon financial and engineering problems where memory and dependence play a crucial role.

Abstract

In this paper, infinite horizon stochastic difference equations and backward stochastic difference equations with fractional noises are studied. The main difficulty comes from fractional noises on infinite horizon. Motivated by discrete-time optimal control problem driven by fractional noises and on infinite horizon, the stochastic maximum principle for discrete-time control problem driven by fractional noises in infinite horizon is proved. As an application, an optimal investment problem is solved.

Maximum principle for optimal control of infinite horizon stochastic difference equations driven by fractional noises

TL;DR

This paper tackles control of infinite-horizon discrete-time systems driven by fractional noises, formulating the problem with state dynamics and a cost represented by where solves a BSE discounted by . It develops an infinite-horizon stochastic maximum principle by proving well-posedness of forward SEs and backward BSEs under tailored weighted norms and establishing adjoint equations, a Hamiltonian, and a verification theorem. The main contributions include a rigorous SMP for fractional-noise-driven systems on infinite horizon, plus an explicit optimal-investment application that yields a computable control law under memory effects from fractional noise. These results extend stochastic control with fractional noise to infinite horizons, enabling robust optimization in long-horizon financial and engineering problems where memory and dependence play a crucial role.

Abstract

In this paper, infinite horizon stochastic difference equations and backward stochastic difference equations with fractional noises are studied. The main difficulty comes from fractional noises on infinite horizon. Motivated by discrete-time optimal control problem driven by fractional noises and on infinite horizon, the stochastic maximum principle for discrete-time control problem driven by fractional noises in infinite horizon is proved. As an application, an optimal investment problem is solved.
Paper Structure (5 sections, 7 theorems, 107 equations, 2 figures)

This paper contains 5 sections, 7 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Infinite horizon S$\Delta$Es and BS$\Delta$Es
  3. A stochastic maximum principle
  4. Application
  5. Acknowledgments

Key Result

Proposition 1

Let $\delta_n=1-(n+2)^{-\theta}$, $\theta>1$. Then $|\vec{\delta}^\theta|\ge exp\left(\frac{2^{1-\theta}}{1-\theta}+\frac{2^{1-2\theta}}{1-2\theta}\right)$.

Figures (2)

  • Figure 1: The case driven by fractional noise with $H=0.75$.
  • Figure 2: The case driven by fractional noise with $H=0.25$.

Theorems & Definitions (18)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Proposition 4
  • proof
  • ...and 8 more