Stochastic control problems for systems driven by normal martingales

TL;DR

This paper investigates stochastic control problems where controlling jump sizes alters the underlying martingale type, establishing theoretical foundations, deriving the HJB equation, and proving solution uniqueness.

Contribution

It introduces a novel class of control problems involving jump size manipulation, providing existence, Bellman principle, HJB equation derivation, and viscosity solution uniqueness.

Findings

Established existence of solutions on Wiener--Poisson space.

Formulated the control problem and derived the HJB equation.

Proved uniqueness of the viscosity solution for the HJB equation.

Abstract

In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener--Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the nonuniqueness for such solutions. Based on these theoretical results, we then formulate the control problem and prove the Bellman principle, and derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally, we prove a uniqueness result for the viscosity solution of such an HJB equation.