A Review of First-Passage Theory for the Segerdahl Risk Process and Extensions

TL;DR

This paper reviews first-passage theory for the Segerdahl risk process, highlighting its unique features, existing methods, and open problems, with an emphasis on computational challenges and connections to other spectrally negative Markov processes.

Contribution

It provides a comprehensive review of approaches to first-passage problems in the Segerdahl process and discusses open issues in numerical computation of key functions.

Findings

The ruin probability for the Segerdahl process has been computed.

Connections between different approaches to first-passage problems are identified.

Open problems in numerical methods for non-Levy, non-diffusion processes are highlighted.

Abstract

The Segerdahl process (Segerdahl (1955)), characterized by exponential claims and affine drift, has drawn a considerable amount of interest -- see, for example, (Tichy (1984); Avram and Usabel (2008), due to its economic interest (it is the simplest risk process which takes into account the effect of interest rates). It is also the simplest non-Levy, non-diffusion example of a spectrally negative Markov risk model. Note that for both spectrally negative Levy and diffusion processes, first passage theories which are based on identifying two basic monotone harmonic functions/martingales have been developped. This means that for these processes many control problems involving dividends, capital injections, etc., may be solved explicitly once the two basic functions have been obtained. Furthermore, extensions to general spectrally negative Markov processes are possible (Landriault et al. (2017), Avram et al. (2018); Avram and Goreac (2019); Avram et al. (2019b). Unfortunately, methods for computing the basic functions are still lacking outside the Levy and diffusion classes, with the notable exception of the Segerdahl process, for which the ruin probability has been computed (Paulsen and Gjessing (1997). However, there is a striking lack of numerical results in both cases. This motivated us to review several approaches, with the purpose of drawing attention to connections between them, and underlying open problems.