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Maximum principle for recursive optimal control problem of stochastic delay evolution equations

Guomin Liu, Jian Song, Meng Wang

Abstract

For a class of stochastic delay evolution equations driven by cylindrical $Q$-Wiener process, we study the Pontryagin's maximum principle for the stochastic recursive optimal control problem. The delays are given as moving averages with respect to general finite measures and appear in all the coefficients of the control system. In particular, the final cost can contain the state delay. To derive the main result, we introduce a new form of anticipated backward stochastic evolution equations with terminals acting on an interval as the adjoint equations of the delayed state equations and deploy a proper dual analysis between them. Under certain convex assumption on the coefficient function and the Hamiltonian, we also show sufficiency of the maximum principle.

Maximum principle for recursive optimal control problem of stochastic delay evolution equations

Abstract

For a class of stochastic delay evolution equations driven by cylindrical -Wiener process, we study the Pontryagin's maximum principle for the stochastic recursive optimal control problem. The delays are given as moving averages with respect to general finite measures and appear in all the coefficients of the control system. In particular, the final cost can contain the state delay. To derive the main result, we introduce a new form of anticipated backward stochastic evolution equations with terminals acting on an interval as the adjoint equations of the delayed state equations and deploy a proper dual analysis between them. Under certain convex assumption on the coefficient function and the Hamiltonian, we also show sufficiency of the maximum principle.
Paper Structure (14 sections, 12 theorems, 153 equations)

This paper contains 14 sections, 12 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SDEEs and anticipated BSEEs
  4. Stochastic delay evolution equations
  5. Anticipated backward stochastic evolution equations
  6. Stochastic maximum principle
  7. Formulation of the control problem
  8. Variational equations
  9. Maximum principle
  10. Sufficient conditions
  11. Some applications
  12. Optimal control problem of SPDEs with delay
  13. LQ problem for SDEEs
  14. Proof of Theorem \ref{['exu']}

Key Result

Lemma 2.1

Suppose that for each $\varphi \in V$, it holds for $dt\times dP$-almost all $(t,\omega )\in \lbrack 0,T]\times \Omega$. Then there exists an adapted càdlàg $H$-valued process $h(\cdot )$ such that

Theorems & Definitions (35)

  • Lemma 2.1
  • proof
  • Remark 3.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Remark 3.2
  • ...and 25 more