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Relationship between MP and DPP for Risk-Sensitive Stochastic Optimal Control Problems: Viscosity Solution Framework

Huanqing Dong, Jingtao Shi

TL;DR

This work addresses how to connect the general maximum principle ($MP$) and dynamic programming principle ($DPP$) for risk-sensitive stochastic control with nonconvex controls, by recasting the problem as a forward–backward SDE with quadratic generators and analyzing it through viscosity solutions. The authors derive precise relations between the adjoint processes $(p,P)$ and the jets of the value function $V$, both in the state variable and in the time variable, via an appropriate generalized Hamiltonian $\mathcal{H}$ and its time-reduced form $\mathcal{H}_1$. The main contributions include extending the $MP$–$DPP$ connection to risk-sensitive settings with quadratic BSDEs in a non-smooth, viscosity framework, and providing detailed proofs (spatial and temporal) using stochastic-Lipschitz BSDE techniques and BMO-martingale estimates. The results are illustrated with examples showing the applicability even when the value function is not smooth, highlighting the framework’s potential for verification and extension to more complex stochastic systems.

Abstract

In this paper, we study the relationship between general maximum principle and dynamic programming principle for risk-sensitive stochastic optimal control problems, where the control domain is not necessarily convex. The original problem is equivalent to a stochastic recursive optimal control problem of a forward-backward system with quadratic generators. Relations among the adjoint processes, the generalized Hamiltonian function and the value function are proved under the framework of viscosity solutions. Some examples are given to illustrate the theoretical results.

Relationship between MP and DPP for Risk-Sensitive Stochastic Optimal Control Problems: Viscosity Solution Framework

TL;DR

This work addresses how to connect the general maximum principle () and dynamic programming principle () for risk-sensitive stochastic control with nonconvex controls, by recasting the problem as a forward–backward SDE with quadratic generators and analyzing it through viscosity solutions. The authors derive precise relations between the adjoint processes and the jets of the value function , both in the state variable and in the time variable, via an appropriate generalized Hamiltonian and its time-reduced form . The main contributions include extending the – connection to risk-sensitive settings with quadratic BSDEs in a non-smooth, viscosity framework, and providing detailed proofs (spatial and temporal) using stochastic-Lipschitz BSDE techniques and BMO-martingale estimates. The results are illustrated with examples showing the applicability even when the value function is not smooth, highlighting the framework’s potential for verification and extension to more complex stochastic systems.

Abstract

In this paper, we study the relationship between general maximum principle and dynamic programming principle for risk-sensitive stochastic optimal control problems, where the control domain is not necessarily convex. The original problem is equivalent to a stochastic recursive optimal control problem of a forward-backward system with quadratic generators. Relations among the adjoint processes, the generalized Hamiltonian function and the value function are proved under the framework of viscosity solutions. Some examples are given to illustrate the theoretical results.
Paper Structure (11 sections, 9 theorems, 125 equations)

This paper contains 11 sections, 9 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. Problem formulation
  4. Equivalent stochastic recursive optimal control problem
  5. Stochastic Lipschitz BSDEs
  6. MP and DPP
  7. Main result
  8. Some examples
  9. Concluding remarks
  10. Proof of Theorem \ref{['relation of MP and DPP in spatial variable']}
  11. Proof of Theorem \ref{['relation of MP and DPP in time variable']}

Key Result

Lemma 2.1

(1) Let $p \in (1, +\infty)$. Then, there is a positive constant $C_p$ such that, for any $\{\mathcal{F}^t_s\}_{s \geq t}$-adapted and real-valued martingales $M$, we have Thus, for any $p \geq 1$, we write simply $BMO$ for $BMO_p$. (2) Denote by $\mathcal{E}(M)$ the Doleans-Dade exponential of a continuous local martingale $M$, that is, $\mathcal{E}(M_s) = \exp\left\{ M_s - \frac{1}{2}\langle M

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.1
  • Lemma 2.4
  • Definition 3.1
  • Remark 3.1
  • Lemma 3.1
  • Remark 3.2
  • ...and 8 more