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Infinite dimensional open-loop linear quadratic stochastic optimal control problems and related games

Guangdong Jing

Abstract

We investigate the linear quadratic stochastic optimal control problems in infinite dimension without Markovian restriction for coefficients. The necessary and sufficient conditions for open-loop optimal controls are presented. We prove the Fréchet differentiable of the cost functional with respect to the control variable, and the Fréchet derivatives are characterized in detail by operators derived from dual analysis, which are proven to be the stationary conditions. Transposition methods are adopted to deal with the adjoint equations. As applications, we employ the results to study open-loop Nash equilibria for two-person stochastic differential games.

Infinite dimensional open-loop linear quadratic stochastic optimal control problems and related games

Abstract

We investigate the linear quadratic stochastic optimal control problems in infinite dimension without Markovian restriction for coefficients. The necessary and sufficient conditions for open-loop optimal controls are presented. We prove the Fréchet differentiable of the cost functional with respect to the control variable, and the Fréchet derivatives are characterized in detail by operators derived from dual analysis, which are proven to be the stationary conditions. Transposition methods are adopted to deal with the adjoint equations. As applications, we employ the results to study open-loop Nash equilibria for two-person stochastic differential games.
Paper Structure (11 sections, 7 theorems, 64 equations)

This paper contains 11 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Notations and assumptions
  3. Notations
  4. Assumptions
  5. Preliminary materials
  6. Stochastic evolution equation
  7. Backward stochastic evolution equation
  8. Operators derived from state processes
  9. Main results for optimal control
  10. Linear quadratic stochastic differential games
  11. Conclusions and discussions

Key Result

Lemma 3.3

Let Condition eqmarorllccl hold and $f(\cdot, 0)\in L_{\mathbb F}^p(\Omega;L^1(0,T;H))$, $\widetilde{f}(\cdot, 0)\in L_{\mathbb F}^{ p}(\Omega;L^2(0,T;H))$ for some $p\ge2$. Then for any $X_0\in L_{\cal F_0}^{ p}(\Omega;H)$, the above Eq. eqmaror admits a unique mild solution $X(\cdot)\in C_{\mathbb

Theorems & Definitions (18)

  • Definition 1.1
  • Remark 2.3
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.5
  • Lemma 3.6
  • Definition 3.7
  • Proposition 4.1
  • proof
  • Theorem 5.1
  • ...and 8 more