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A global stochastic maximum principle for delayed forward-backward stochastic control systems

Feng Li

Abstract

In this paper, we study a delayed forward-backward stochastic control system in which all the coefficients depend on the state and control terms, and the control domain is not necessarily convex. A global stochastic maximum principle is obtained by using a new method. More precisely, this method introduces first-order and second-order auxiliary equations and offers a novel approach to deriving the adjoint equations as well as the variational equation for $y^\e - y^*$.

A global stochastic maximum principle for delayed forward-backward stochastic control systems

Abstract

In this paper, we study a delayed forward-backward stochastic control system in which all the coefficients depend on the state and control terms, and the control domain is not necessarily convex. A global stochastic maximum principle is obtained by using a new method. More precisely, this method introduces first-order and second-order auxiliary equations and offers a novel approach to deriving the adjoint equations as well as the variational equation for .
Paper Structure (10 sections, 18 theorems, 144 equations)

This paper contains 10 sections, 18 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic maximum principle for delayed forward-backward stochastic control systems
  4. The variational equations of $x^\varepsilon - x^*$
  5. The adjoint equations and the variational equation of $y^\varepsilon - y^*$
  6. The duality relationship
  7. The estimate of $y^\varepsilon(0) - y^*(0)-\hat{y}(0)$
  8. Maximum principle
  9. Stochastic maximum principle for forward-backward stochastic control systems
  10. Appendix

Key Result

Proposition 2.1

Assume that for $\beta >1$, $(\xi, g_1(s,0,0)) \in L^{\beta}_{\mathscr{F}_{T}}(\Omega;\mathbb{R}^m)\times \mathcal{M}_{\mathbb{F}}^{1, \beta}(0,T; \mathbb{R}^m)$ and $g_1 = g_1(t, \omega, \tilde{y}, \tilde{z}): [0, T] \times \Omega \times \mathbb{R}^m \times \mathbb{R}^{m\times d} \rightarrow \mathb has a unique solution $(\tilde{y}, \tilde{z}) \in S^{\beta}_{\mathbb{F}}(0,T;\mathbb{R}^m)\times \

Theorems & Definitions (35)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Lemma 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 25 more