Approximation of optimal ergodic dividend strategies using controlled Markov chains

• Published in: IET control theory & applications Vol. 12; no. 16; pp. 2194 - 2204
• Main Authors: Jin, Zhuo, Yang, Hailiang, Yin, George
• Format: Journal Article
• Language: English
• Published: The Institution of Engineering and Technology 06.11.2018
• Subjects: approximation theory; continuous time systems; controlled Markov chains; discrete time systems; discrete‐time controlled Markov chain; dynamic programming; dynamic programming principle; finite‐time continuous‐time Markov chain; Hamilton–Jacobi–Bellman equations; invariant measure; long‐term average dividend payment; Markov chain approximation techniques; Markov processes; numerical method; optimal control; optimal ergodic dividend payment strategy; optimal ergodic dividend strategies; optimal value; regime switching; regime‐switching model; regime‐switching process subject; Research Article; surplus process; approximation theory; surplus process; regime switching; discrete-time controlled Markov chain; discrete time systems; dynamic programming; optimal control; finite-time continuous-time Markov chain; regime-switching model; Hamilton–Jacobi–Bellman equations; optimal ergodic dividend strategies; optimal ergodic dividend payment strategy; dynamic programming principle; optimal value; numerical method; controlled Markov chains; Markov processes; invariant measure; regime-switching process subject; long-term average dividend payment; Markov chain approximation techniques; continuous time systems
• ISSN: 1751-8644; 1751-8652
• DOI: 10.1049/iet-cta.2018.5394

This study develops a numerical method to find optimal ergodic (long-run average) dividend strategies in a regime-switching model. The surplus process is modelled by a regime-switching process subject to liability constraints. The regime-switching process is modelled by a finite-time continuous-time Markov chain. Using the dynamic programming principle, the optimal long-term average dividend payment is a solution to the coupled system of Hamilton–Jacobi–Bellman equations. Under suitable conditions, the optimal value of the long-term average dividend payment can be determined by using an invariant measure. However, due to the regime switching, getting the invariant measure is very difficult. The objective is to design a numerical algorithm to approximate the optimal ergodic dividend payment strategy. By using the Markov chain approximation techniques, the authors construct a discrete-time controlled Markov chain for the approximation, and prove the convergence of the approximating sequences. A numerical example is presented to demonstrate the applicability of the algorithm.