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Maximum principle for optimal control of stochastic evolution equations with recursive utilities

Guomin Liu, Shanjian Tang

Abstract

We consider the optimal control problem of stochastic evolution equations in a Hilbert space under a recursive utility, which is described as the solution of a backward stochastic differential equation (BSDE). A very general maximum principle is given for the optimal control, allowing the control domain not to be convex and the generator of the BSDE to vary with the second unknown variable $z$. The associated second-order adjoint process is characterized as a unique solution of a conditionally expected operator-valued backward stochastic integral equation.

Maximum principle for optimal control of stochastic evolution equations with recursive utilities

Abstract

We consider the optimal control problem of stochastic evolution equations in a Hilbert space under a recursive utility, which is described as the solution of a backward stochastic differential equation (BSDE). A very general maximum principle is given for the optimal control, allowing the control domain not to be convex and the generator of the BSDE to vary with the second unknown variable . The associated second-order adjoint process is characterized as a unique solution of a conditionally expected operator-valued backward stochastic integral equation.
Paper Structure (19 sections, 16 theorems, 264 equations)

This paper contains 19 sections, 16 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Conditionally expected operator-valued BSIEs
  3. Conditional expectation for operator-valued random variables
  4. Existence of the conditional expectation
  5. An aggregation theorem and proof of Theorem \ref{['Myth2-17']}
  6. Formulation of the BSIE
  7. Existence and uniqueness of solutions
  8. Itô's formula
  9. Evolution operators associated to forward SEEs
  10. Itô's formula in a weak formulation
  11. Stochastic maximum principle for optimally controlled SEEs
  12. Formulation of the problem
  13. Adjoint equations and the maximum principle
  14. Proof of Theorem \ref{['SMP']}
  15. Application on controlled SPDEs
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let $Y\in\mathfrak{L}_{2}(H\times H;L^{1}(\mathcal{F}))$. Then the conditional expectation $\mathbb{E}[Y|\mathcal{G}]$ exists and is integrable (i.e., $\mathbb{E}[Y|\mathcal{G}]\in L_{w}^{1}(\mathcal{G},\mathfrak{L}_{2}(H\times H))$) if and only if the mapping $(u,v)\longmapsto \mathbb{E}[Y(u,v)|\ma for some $0\leq g\in L^{1}(\mathcal{G})$. Moreover, such an $\mathbb{E}[Y|\mathcal{G}]$ is unique (

Theorems & Definitions (36)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 26 more