Solving Optimal Dividend Problems via Phase-type Fitting Approximation of Scale Functions

TL;DR

This paper develops a phase-type fitting approximation method for solving optimal dividend problems in spectrally negative Lévy models, enabling efficient computation of scale functions for complex processes.

Contribution

It introduces a novel approximation approach using phase-type and meromorphic Lévy processes to estimate scale functions for general spectrally negative Lévy processes.

Findings

The approximation converges analytically for various Lévy processes.

Numerical examples demonstrate high accuracy of the method.

Effective for processes with Weibull, β-family, and CGMY jumps.

Abstract

The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative L\'evy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We consider spectrally negative L\'evy processes with phase-type jumps as well as meromorphic L\'evy processes (Kuznetsov et al., 2010a), and use their scale functions to approximate the scale function for a general spectrally negative L\'evy process. We obtain analytically the convergence results and illustrate numerically the effectiveness of the approximation methods using examples with the spectrally negative L\'evy process with i.i.d. Weibull-distributed jumps, the \beta-family and CGMY process.