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Anticipated backward stochastic evolution equations and maximum principle for path-dependent systems in infinite dimensions

Guomin Liu, Jian Song, Meng Wang

Abstract

For a class of path-dependent stochastic evolution equations driven by cylindrical $Q$-Wiener process, we study the Pontryagin's maximum principle for the stochastic recursive optimal control problem. In this infinite-dimensional control system, the state process depends on its past trajectory, the control is delayed via an integral with respect to a general finite measure, and the final cost relies on the delayed state.To obtain the maximum principle, we introduce a functional adjoint operator for the non-anticipative path derivative and establish the well-posedness of an anticipated backward stochastic evolution equation in the path-dependent form, which serves as the adjoint equation.

Anticipated backward stochastic evolution equations and maximum principle for path-dependent systems in infinite dimensions

Abstract

For a class of path-dependent stochastic evolution equations driven by cylindrical -Wiener process, we study the Pontryagin's maximum principle for the stochastic recursive optimal control problem. In this infinite-dimensional control system, the state process depends on its past trajectory, the control is delayed via an integral with respect to a general finite measure, and the final cost relies on the delayed state.To obtain the maximum principle, we introduce a functional adjoint operator for the non-anticipative path derivative and establish the well-posedness of an anticipated backward stochastic evolution equation in the path-dependent form, which serves as the adjoint equation.

Paper Structure

This paper contains 14 sections, 14 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PSEEs and anticipated BSEEs
  4. Path-dependent stochastic evolution equations
  5. Anticipated backward stochastic evolution equations
  6. Path derivative and its adjoint operator
  7. Stochastic maximum principle
  8. Formulation of the control problem
  9. Variational equations
  10. Maximum principle
  11. Sufficient conditions
  12. Some applications
  13. Optimal control problem of path-dependent SPDEs
  14. LQ problem for PSEEs

Key Result

Lemma 2.1

Suppose that for each $\varphi \in V$, it holds that 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 (46)

  • Lemma 2.1
  • proof
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 36 more