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Reflected stochastic recursive control problems with jumps: dynamic programming and stochastic verification theorems

Lu Liu, Qingmeng Wei

TL;DR

The paper develops a framework for reflected stochastic recursive control with jumps, where the state obeys a jump–diffusion SDE and the cost is given by the solution to a reflected BSDE with jumps. By establishing a dynamic programming principle via a backward stochastic semigroup, it connects the value function $W(t,x)$ to a HJB obstacle PIDE in the viscosity sense and constructs penalized problems $W^n$ that converge to $W$, ensuring existence and uniqueness of the viscosity solution. It also proves key regularity properties of $W$, notably joint Lipschitz continuity and, under suitable conditions, semiconcavity in $(t,x)$, using Kulik’s time transformation to handle jumps. These regularity results underpin stochastic verification theorems and allow the derivation of optimal feedback controls from viscosity solutions, while relaxing prior restrictions such as reflection freezing and driver diffusion independence.

Abstract

This paper mainly investigates reflected stochastic recursive control problems governed by jump-diffusion dynamics. The system's state evolution is described by a stochastic differential equation driven by both Brownian motion and Poisson random measures, while the recursive cost functional is formulated via the solution process Y of a reflected backward stochastic differential equation driven by the same dual stochastic sources. By establishing the dynamic programming principle, we provide the probabilistic interpretation of an obstacle problem for partial integro-differential equations of Hamilton-Jacobi-Bellman type in the viscosity solution sense through our control problem's value function. Furthermore, the value function is proved to inherit the semi-concavity and joint Lipschitz continuity in state and time coordinates, which play key roles in deriving stochastic verification theorems of control problem within the framework of viscosity solutions. We remark that some restrictions in previous study are eliminated, such as the frozen of the reflected processes in time and state, and the independence of the driver from diffusion variables.

Reflected stochastic recursive control problems with jumps: dynamic programming and stochastic verification theorems

TL;DR

The paper develops a framework for reflected stochastic recursive control with jumps, where the state obeys a jump–diffusion SDE and the cost is given by the solution to a reflected BSDE with jumps. By establishing a dynamic programming principle via a backward stochastic semigroup, it connects the value function to a HJB obstacle PIDE in the viscosity sense and constructs penalized problems that converge to , ensuring existence and uniqueness of the viscosity solution. It also proves key regularity properties of , notably joint Lipschitz continuity and, under suitable conditions, semiconcavity in , using Kulik’s time transformation to handle jumps. These regularity results underpin stochastic verification theorems and allow the derivation of optimal feedback controls from viscosity solutions, while relaxing prior restrictions such as reflection freezing and driver diffusion independence.

Abstract

This paper mainly investigates reflected stochastic recursive control problems governed by jump-diffusion dynamics. The system's state evolution is described by a stochastic differential equation driven by both Brownian motion and Poisson random measures, while the recursive cost functional is formulated via the solution process Y of a reflected backward stochastic differential equation driven by the same dual stochastic sources. By establishing the dynamic programming principle, we provide the probabilistic interpretation of an obstacle problem for partial integro-differential equations of Hamilton-Jacobi-Bellman type in the viscosity solution sense through our control problem's value function. Furthermore, the value function is proved to inherit the semi-concavity and joint Lipschitz continuity in state and time coordinates, which play key roles in deriving stochastic verification theorems of control problem within the framework of viscosity solutions. We remark that some restrictions in previous study are eliminated, such as the frozen of the reflected processes in time and state, and the independence of the driver from diffusion variables.
Paper Structure (8 sections, 25 theorems, 140 equations, 1 figure)

This paper contains 8 sections, 25 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reflected backward stochastic differential equations with jumps
  4. Reflected stochastic recursive control problems with jumps
  5. Formulation of the control problem
  6. The Obstacle Problems of partial integral-differential equations of HJB type
  7. More properties of the value function
  8. The semi-concavity of $W(\cdot,\cdot)$ in $(t,x)$

Key Result

Lemma 2.1

Assume (A$_1$) and (A$_2$) hold. Then the following two results hold true. (i) The sequence $\{(Y^n, Z^n, V^n, A^n)\}_{n\geqslant 0}$ has a limit $(Y,Z,V,A)$ such that $Y^n$ converges decreasingly to $Y \in \mathcal{S}^2_{ \mathbb F }( 0,T; \mathbb R )$, and $(Z,V,A)$ is the weak limit of $(Z^n,V^n

Figures (1)

  • Figure 1: Relations among the auxiliary Lemmas

Theorems & Definitions (40)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.1
  • Proposition 3.3
  • Proposition 3.4
  • Definition 3.2
  • ...and 30 more