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The Optimal Control Problem of Stochastic Differential System with Extended Mixed Delays and Applications

Xinpo Li, Jingtao Shi

TL;DR

This work develops a comprehensive framework for stochastic optimal control of systems with extended mixed delays, incorporating point delays, extended distributed delays, and extended noisy memory in both state and control. By transforming delay variational equations into Volterra integral equations without delays and employing a coefficient-decomposition approach together with Malliavin calculus, the authors derive a stochastic maximum principle and a verification theorem; the adjoint system is recast via Clark-Ocone to ABSDEs, linking to forward-backward integral equation theory. The paper also presents a nonzero-sum stochastic differential game with extended delays and a solvable linear-quadratic example to illustrate practical computation of optimal strategies. These results expand the toolkit for memory- and delay-laden stochastic dynamics, with potential applications in finance and engineering where historical dependence and stochastic memory effects are essential.

Abstract

This paper investigates an optimal control problem where the system is described by a stochastic differential equation with extended mixed delays that contain point delay, extended distributed delay, and extended noisy memory. The model is general in that the extended mixed delays of the state variable and control variable are components of all the coefficients, in particular, the diffusion term and the terminal cost. To address the difficulties induced by the extended noisy memory, by stochastic Fubini theorem, we transform the delay variational equation into a Volterra integral equation without delay, and then a kind of backward stochastic Volterra integral equation with Malliavin derivatives is introduced by the developed coefficient decomposition method and the generalized duality principle. Therefore, the stochastic maximum principle and the verification theorem are established. Subsequently, with Clark-Ocone formula, the adjoint equation is expressed as a set of anticipated backward stochastic differential equations. Finally, a nonzero-sum stochastic differential game with extended mixed delays and a linear-quadratic solvable example are discussed, as applications.

The Optimal Control Problem of Stochastic Differential System with Extended Mixed Delays and Applications

TL;DR

This work develops a comprehensive framework for stochastic optimal control of systems with extended mixed delays, incorporating point delays, extended distributed delays, and extended noisy memory in both state and control. By transforming delay variational equations into Volterra integral equations without delays and employing a coefficient-decomposition approach together with Malliavin calculus, the authors derive a stochastic maximum principle and a verification theorem; the adjoint system is recast via Clark-Ocone to ABSDEs, linking to forward-backward integral equation theory. The paper also presents a nonzero-sum stochastic differential game with extended delays and a solvable linear-quadratic example to illustrate practical computation of optimal strategies. These results expand the toolkit for memory- and delay-laden stochastic dynamics, with potential applications in finance and engineering where historical dependence and stochastic memory effects are essential.

Abstract

This paper investigates an optimal control problem where the system is described by a stochastic differential equation with extended mixed delays that contain point delay, extended distributed delay, and extended noisy memory. The model is general in that the extended mixed delays of the state variable and control variable are components of all the coefficients, in particular, the diffusion term and the terminal cost. To address the difficulties induced by the extended noisy memory, by stochastic Fubini theorem, we transform the delay variational equation into a Volterra integral equation without delay, and then a kind of backward stochastic Volterra integral equation with Malliavin derivatives is introduced by the developed coefficient decomposition method and the generalized duality principle. Therefore, the stochastic maximum principle and the verification theorem are established. Subsequently, with Clark-Ocone formula, the adjoint equation is expressed as a set of anticipated backward stochastic differential equations. Finally, a nonzero-sum stochastic differential game with extended mixed delays and a linear-quadratic solvable example are discussed, as applications.
Paper Structure (11 sections, 13 theorems, 136 equations)

This paper contains 11 sections, 13 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. Transform SDDE into SVIE
  4. Stochastic optimal control problem with extended mixed delays
  5. Stochastic maximum principle
  6. Verification Theorem
  7. Transform BSVIE into ABSDE
  8. Applications
  9. Nonzero-sum stochastic differential game with extended mixed delays
  10. A solvable LQ example
  11. Concluding remarks

Key Result

Proposition 2.1

Assuming (H2.1) and (H2.2) are satisfied, let $\tilde{\xi}(\cdot):\Omega\to C([t_0-\delta,t_0]; \mathbb{R}^n)$ be $\mathcal{F}_{t_0}$-measurable and $\mathbb{E}[\sup_{t\in [t_0-\delta,t_0]}|\tilde{\xi}(t)|^2]<\infty$, with $\tilde{\phi}(\cdot,\cdot),\tilde{\psi}(\cdot,\cdot)\in L_{\mathcal{F}_t}^{\i

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 4.1
  • ...and 14 more