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Maximum Principles for Partially Observed Controls of Forward SPDEs and Backward SDEs with Jumps

Hongjiang Qian, George Yin, Yanzhao Cao, Guannan Zhang

Abstract

This work establishes two versions of the Pontryagin-type maximum principles for partially observed optimal control of coupled forward stochastic partial differential equations (FSPDEs) and backward stochastic differential equations (BSDEs) with jumps in convex control domains. The FSPDE-BSDE system is driven by cylindrical Wiener processes, finite-dimensional Brownian motions, and compensated Poisson random measures. For systems with deterministic coefficients, a direct method is employed and particular attention is focused on establishing the well-posedness of a singular backward SPDE with jumps. For systems with random coefficients, a Malliavin calculus approach is developed. The main novelty here is the establishment of the well-posedness of an operator-valued SPDE with jumps, which provides a new stochastic flow representation for linear SPDEs with jumps.

Maximum Principles for Partially Observed Controls of Forward SPDEs and Backward SDEs with Jumps

Abstract

This work establishes two versions of the Pontryagin-type maximum principles for partially observed optimal control of coupled forward stochastic partial differential equations (FSPDEs) and backward stochastic differential equations (BSDEs) with jumps in convex control domains. The FSPDE-BSDE system is driven by cylindrical Wiener processes, finite-dimensional Brownian motions, and compensated Poisson random measures. For systems with deterministic coefficients, a direct method is employed and particular attention is focused on establishing the well-posedness of a singular backward SPDE with jumps. For systems with random coefficients, a Malliavin calculus approach is developed. The main novelty here is the establishment of the well-posedness of an operator-valued SPDE with jumps, which provides a new stochastic flow representation for linear SPDEs with jumps.
Paper Structure (15 sections, 19 theorems, 108 equations)

This paper contains 15 sections, 19 theorems, 108 equations.

Table of Contents

  1. Motivation
  2. Prelude and notation
  3. Formulation
  4. Direct approach
  5. First variations
  6. Adjoint equations
  7. Stochastic maximum principle
  8. Singular backward SPDEs with jumps
  9. Linear FSPDEs with jumps and auxiliary estimates
  10. Existence of singular BSPDEs with jumps
  11. Trace class regularity of $Q_{1}$
  12. Malliavin calculus approach
  13. Preliminaries for Malliavin calculus
  14. Stochastic flows for linear SPDEs with jumps
  15. Stochastic maximum principles

Key Result

Lemma 3.1

Let Assumptions ass holds. For any $u\in \mathcal{U}_{\text{ad}}$, there exists a constant $C>0$ such that the solution of sys1 and eq-rho satisfy the following estimates: $\forall\, \kappa \geq 2$, we have ${\mathbb E}^u |\rho^u_t|^\kappa <\infty$, and

Theorems & Definitions (41)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark 3.6
  • Definition 3.7
  • ...and 31 more