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Transposition Approach to Optimal Control of McKean-Vlasov SPDEs

Liangying Chen, Wilhelm Stannat

Abstract

In this paper, we investigate an optimal control problem for McKean-Vlasov stochastic partial differential equations, in which the coefficients depend on the law of the state process. For systems with nonconvex control sets, we establish a Pontryagin-type stochastic maximum principle that provides necessary optimality conditions for admissible controls. The analysis is based on the classical spike variation method together with the introduction of an adjoint backward stochastic partial differential equation involving Lions derivatives with respect to probability measures. Our results extend the stochastic maximum principle for McKean-Vlasov controlled stochastic differential equations to the infinite-dimensional SPDE setting.

Transposition Approach to Optimal Control of McKean-Vlasov SPDEs

Abstract

In this paper, we investigate an optimal control problem for McKean-Vlasov stochastic partial differential equations, in which the coefficients depend on the law of the state process. For systems with nonconvex control sets, we establish a Pontryagin-type stochastic maximum principle that provides necessary optimality conditions for admissible controls. The analysis is based on the classical spike variation method together with the introduction of an adjoint backward stochastic partial differential equation involving Lions derivatives with respect to probability measures. Our results extend the stochastic maximum principle for McKean-Vlasov controlled stochastic differential equations to the infinite-dimensional SPDE setting.
Paper Structure (10 sections, 9 theorems, 122 equations)

This paper contains 10 sections, 9 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Mathematical setting and main result
  3. Lions Derivative in Infinite Dimension
  4. Mathematical Setting
  5. Main Result
  6. Variational Equations
  7. Well-Posedness within the Transposition Solution Framework
  8. Transposition Solution to an H-valued McKean-Vlasov BSEE
  9. Regularity of second-order adjoint equation
  10. Proof of Pontryagin Maximum Principle

Key Result

Lemma 2.1

If $\varphi: H\times \mathcal{P}_2 (H)\to E$ is continuously Fréchet differentiable in its first component and $\Lambda$-continuously Fréchet differentiable in its second component, then there exists a jointly measurable version of

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.1
  • Remark 2.1
  • Theorem 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.1
  • ...and 6 more