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A General Maximum Principle for Progressive Optimal Control of Fully Coupled Forward-Backward Stochastic Systems with Jumps

Bin Wang, Yu Si, Jingtao Shi

TL;DR

This work addresses stochastic control problems driven by fully coupled forward-backward SDEs with jumps, allowing nonconvex control domains and a diffusion term that depends on the jump variable $e$. It develops a general maximum principle using spike variation and a pair of adjoint BSDEs, with a Hamiltonian that incorporates the jump structure and $e$-dependent $Z$-term, and proves an inequality that the optimal control must satisfy for all admissible controls. The authors establish $L^p$-estimates and well-posedness for fully coupled FBSDEPs with jumps under their assumptions, enabling rigorous derivation of the maximum principle. The results extend classical maximum principles to nonconvex controls and jump-driven systems, broadening the applicability to areas such as recursive utility in finance and economics where jumps and $Z$-dependence on $e$ are essential. The framework unifies and extends prior theories, providing a foundation for further exploration of optimal control in systems with both continuous fluctuations and abrupt events.

Abstract

This paper is concerned with a general maximum principle for the fully coupled forward-backward stochastic optimal control problem with jumps, where the control domain is not necessarily convex, within the progressively measurable framework. A distinct feature in this paper is that the solution $Z$ of BSDEPs could include the variable ``$e$'', further, the diffusion term of BSDEPs takes the form $\int_{\mathcal{E}}Z_{(t,e)}ν(d e)d W_t$ rather than the conventional $Z_t dW_t$, reflecting the essential coupling between the solution component $Z$ and the Polish space $\mathcal{E}$.

A General Maximum Principle for Progressive Optimal Control of Fully Coupled Forward-Backward Stochastic Systems with Jumps

TL;DR

This work addresses stochastic control problems driven by fully coupled forward-backward SDEs with jumps, allowing nonconvex control domains and a diffusion term that depends on the jump variable . It develops a general maximum principle using spike variation and a pair of adjoint BSDEs, with a Hamiltonian that incorporates the jump structure and -dependent -term, and proves an inequality that the optimal control must satisfy for all admissible controls. The authors establish -estimates and well-posedness for fully coupled FBSDEPs with jumps under their assumptions, enabling rigorous derivation of the maximum principle. The results extend classical maximum principles to nonconvex controls and jump-driven systems, broadening the applicability to areas such as recursive utility in finance and economics where jumps and -dependence on are essential. The framework unifies and extends prior theories, providing a foundation for further exploration of optimal control in systems with both continuous fluctuations and abrupt events.

Abstract

This paper is concerned with a general maximum principle for the fully coupled forward-backward stochastic optimal control problem with jumps, where the control domain is not necessarily convex, within the progressively measurable framework. A distinct feature in this paper is that the solution of BSDEPs could include the variable ``'', further, the diffusion term of BSDEPs takes the form rather than the conventional , reflecting the essential coupling between the solution component and the Polish space .
Paper Structure (10 sections, 7 theorems, 105 equations)

This paper contains 10 sections, 7 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and preliminaries
  3. Problem formulation
  4. $L^p$-estimate of fully coupled FBSDEPs
  5. General maximum principle
  6. Concluding remarks
  7. Proof of Proposition 2.1
  8. Proof of Lemma 3.1
  9. Proof of Lemma 3.2
  10. Proof of Lemma 3.3

Key Result

Lemma 2.1

For any stochastic processes $\left(y_\cdot,z_{(\cdot,\cdot)},\tilde{z}_{(\cdot,\cdot)}\right)\in \mathscr{N}^p[0,T]$, consider the following FBSDEP: where $b,\sigma,g:[0,T]\times\mathbf{R}\times\mathbf{R}\times\mathcal{L}^2\times\mathcal{L}^2\times\mathbf{U}\rightarrow\mathbf{R}$ , $f:[0,T]\times\mathbf{R}\times\mathbf{R}\times\mathcal{L}^2\times\mathcal{L}^2\times\mathbf{U}\times\mathcal{E}\rig

Theorems & Definitions (18)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Remark 2.2
  • Lemma 3.1
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.2
  • Lemma 3.3
  • ...and 8 more