Weak Necessary and Sufficient Stochastic Maximum Principle for Markovian Regime-Switching Diffusion Models

TL;DR

This paper establishes a weak maximum principle for stochastic control problems with regime switching, using Clarke's generalized gradient, and provides conditions under which it is also sufficient, supported by four examples.

Contribution

It introduces a novel weak maximum principle based on Clarke's calculus for Markovian regime-switching models, relaxing traditional maximum conditions.

Findings

The maximum principle is both necessary and sufficient under certain convexity conditions.

Four examples demonstrate the application of the weak maximum principle.

The approach broadens the scope of stochastic control theory for regime-switching models.

Abstract

In this paper we prove a weak necessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.