A Risk Model with an Observer in a Markov Environment

Abstract

We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory, we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration.

Lévy insurance risk process with Poissonian taxation

“…n = 1) is known to result in nice explicit formulas (see Albrecher, Cheung et al (2011, Sections 2 and 4.1); Albrecher et al (2013, Section 2)). Since then, ruin theory under a Poissonian observer has been further developed by Albrecher & Ivanovs (2013) and Albrecher et al (2015), who looked at a Markov additive risk process and a Lévy risk process, respectively. Indeed, exponential inter-observation times are also related to the case of constant bankruptcy rate in the (Gamma-)Omega risk model.…”

Spectrally negative Lévy processes with Parisian reflection below and classical reflection above

“…(capital injections) Capital injections correspond to Parisian reflection below. We compute their total discounted values for both (1) and (2) for the infinite horizon case as well as for the cases that are killed upon exiting [a, b], [a, ∞) and (−∞, b] for a < 0 < b.…”

“…(dividends) If dividends are assumed to be paid continuously, then they are modeled by the classical reflection above in process (2). We compute their total expected discounted values for the infinite horizon case and for the case that is killed upon exiting [a, ∞) for a < 0.…”

“…The scale functions and their applications are also reviewed. Section 3 presents the main results for both processes (1) and (2) and their corollaries. Sections 4 and 5 give the proofs of the main results related to process (1) for the bounded and unbounded variation cases, respectively.…”

“…Sections 4 and 5 give the proofs of the main results related to process (1) for the bounded and unbounded variation cases, respectively. Finally, Section 6 gives those of the theorems related to (2). Some proofs of the corollaries are provided in the appendix.…”