On a mixed singular/switching control problem with multiples regimes

TL;DR

This paper investigates a complex stochastic control problem involving both singular and switching controls across multiple regimes, establishing the connection between the value function and a Hamilton-Jacobi-Bellman equation with specific regularity properties.

Contribution

It introduces a novel analysis of a mixed singular/switching control problem using probabilistic and PDE methods, proving the value function's regularity and its characterization via HJB equations.

Findings

Value function matches the solution of a HJB equation.

Value function has regularity $C^{0,1}igcap ext{W}^{2, ext{infty}}_{loc}$.

Method combines probabilistic, PDE, and penalization techniques.

Abstract

This paper studies {a} mixed singular/switching stochastic control problem for a multidimensional diffusion with multiples regimes on a bounded domain. Using probabilistic, partial differential equation (PDE) and penalization techniques, we show that the value function associated with this problem agrees with the solution to a Hamilton-Jacobi-Bellman (HJB) equation. In that way, we see that the regularity of the value function is $C^{0,1}\cap\W^{2,\infty}_{loc}$.