Time-inconsistent optimal control problems with regime-switching

Jiaqin Wei

doi:10.3934/mcrf.2017022

Article Contents

2017, Volume 7, Issue 4: 585-622. Doi: 10.3934/mcrf.2017022

This issue Previous Article A stochastic control problem and related free boundaries in finance Next Article Addendum to "A sparse Markov chain approximation of LQ-type stochastic control problems"

Time-inconsistent optimal control problems with regime-switching

Jiaqin Wei

School of Statistics, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China

Received: December 31, 2015

Revised: March 31, 2017

Published: November 2017

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Abstract

In this paper, a time-inconsistent optimal control problem is studied for diffusion processes modulated by a continuous-time Markov chain. In the performance functional, the running cost and terminal cost depend on not only the initial time, but also the initial state of the Markov chain. By modifying the method of multi-person game, we obtain an equilibrium Hamilton-Jacobi-Bellman equation under proper conditions. The well-posedness of this equilibrium HJB Equation is studied in the case where the diffusion term is independent of the control variable. Furthermore, a time-inconsistent linear-quadratic control problem is considered as a special case.

Keywords:

Time-inconsistence,
regime-switching,
multi-person game,
equilibrium Hamilton-Jacobi-Bellman equation,
linear-quadratic control.

Mathematics Subject Classification: Primary: 93E20, 49L20, 49N10, 49N70; Secondary: 35Q93, 91A23, 91A65.

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Figure 1. The solutions for $P(1,t,1)$ and $P(2,t,2)$

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