A stochastic control approach for constrained stochastic differential games with jumps and regimes

TL;DR

This paper introduces a stochastic control framework for constrained stochastic differential games with jumps and regime switching, linking optimal control and Lagrangian methods, and illustrating with a Bancassurance example.

Contribution

It develops a novel approach for constrained stochastic differential games with jumps and regimes, incorporating stochastic Lagrange multipliers and applying stochastic maximum principle techniques.

Findings

Established relations between stochastic control and Lagrangian methods.

Proved theorems for different types of constraints with Lagrange multipliers.

Applied the theory to a Bancassurance cooperation model.

Abstract

We develop an approach for two player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different type of constraints, which lead us to find real valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with stochastic maximum principle technique. Our application is an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type, called Bancassurance.