  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.
