A free boundary problem arising from a multi-state regime-switching stock trading model

TL;DR

This paper analyzes a free boundary problem from a multi-state regime-switching stock trading model, proving boundary properties and deriving optimal trading strategies in a complex stochastic setting.

Contribution

It establishes the smoothness, non-overlapping, and monotonicity of four switching free boundaries and fully determines their relative positions, advancing understanding of regime-switching trading models.

Findings

All four switching free boundaries are non-overlapping, monotonic, and smooth.

The relative positions of the free boundaries are fully characterized.

Optimal trading strategies are derived from the boundary analysis.

Abstract

In this paper, we study a free boundary problem, which arises from an optimal trading problem of a stock that is driven by a uncertain market status process. The free boundary problem is a variational inequality system of three functions with a degenerate operator. The main contribution of this paper is that we not only prove all the four switching free boundaries are no-overlapping, monotonic and $C^{\infty}$-smooth, but also completely determine their relative localities and provide the optimal trading strategies for the stock trading problem.