Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections

• Published in: Automatica (Oxford) Vol. 49; no. 8; pp. 2317 - 2329
• Main Authors: Jin, Zhuo, Yang, Hailiang, George Yin, G.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01-08-2013; Elsevier
• Subjects: Applied sciences; Approximation; Calculus of variations and optimal control; Capital injection; Control equipment; Diffusion; Dividend policy; Exact sciences and technology; Firm modelling; Free boundary; Inference from stochastic processes; time series analysis; Investment strategy; Markov chain approximation; Markov chains; Mathematical analysis; Mathematical models; Mathematical programming; Mathematics; Numerical analysis; Operational research and scientific management; Operational research. Management science; Optimization; Probability and statistics; Sciences and techniques of general use; Singular control; Statistics; Stochastic control; Singular control; Dividend policy; Free boundary; Markov chain approximation; Investment strategy; Stochastic control; Capital injection; Approximation algorithm; Modeling; Surplus; Private equity; Markov chain; Variational inequality; Diffusion process; Jump process; Dynamic programming; Integrodifferential equation; State constraint; Dividend; Time series; Non linear system; Exact solution; Optimal control; regime switching model; Discrete time; Investment
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2013.04.043

This work focuses on numerical methods for finding optimal investment, dividend payment, and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections. The surplus is modeled by a regime-switching jump diffusion process subject to both regular and singular controls. Using the dynamic programming principle, the value function is a solution of the coupled system of nonlinear integro-differential quasi-variational inequalities. In this paper, the state constraint of the impulsive control gives rise to a capital injection region with free boundary, which makes the problem even more difficult to analyze. Together with the regular control and regime-switching, the closed-form solutions are virtually impossible to obtain. We use Markov chain approximation techniques to construct a discrete-time controlled Markov chain to approximate the value function and optimal controls. Convergence of the approximation algorithms is proved. Examples are presented to illustrate the applicability of the numerical methods.