This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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.

In classical risk theory, the ruin of an insurance portfolio is defined as the event that the surplus process becomes negative. In practice, it may be more reasonable to assume that the surplus value is not checked continuously, but at certain times only. If these times are not fixed deterministically, but are assumed to be epochs of a certain independent renewal process, then one often still has sufficient analytical structure to obtain explicit expressions for ruin probabilities and related quantities; see [

In this paper, we extend the above model and allow the surplus process to be a spectrally-negative Markov additive process. The dynamics of such a process change according to an external environment process, modeled by a Markov chain, and changes of the latter may also cause a jump in the surplus process. We assume that the value of the surplus process is only observed at epochs of a Poisson process, and ruin occurs when at any such observation time, the surplus process is negative. We also allow the rate of observations to depend on the current state of the environment (one possible interpretation being that if the environment states refer to different economic conditions, a regulator may increase the observation rates in states of distress). Using an approximation argument and the spectral theory for Markov additive processes, we explicitly calculate for any initial capital the survival probability and the probability to reach a given level before ruin in this model. The resulting formulas turn out to be quite simple. At the same time, these formulas provide information on certain occupation times of the process, which may be of independent theoretical interest.

In

Let

Write

As in [

It is known that

Let us quickly recall the recently established exit theory for spectrally-negative MAPs, which is an extension of the one for scalar Lévy processes (see, e.g., [

The two-sided exit problem for MAPs without positive jumps was solved in [

Importantly, all the above identities hold for defective (killed) MAPs, as well, i.e., when the state space of

Note that killed MAPs preserve the stationarity and independence of increments given the environment state. Furthermore, we get probabilistic identities of the following type:

Letting

The following main result determines the matrix of probabilities of reaching a level,

The vector of survival probabilities according to our relaxed ruin concept has the following simple form:

Equation (

Equation (

The proofs rely on a spectral representation of the matrix,

According to Equation (

The proof of Theorem 4.1 relies on an approximation idea, which has already appeared in various papers; see, e.g., [

By the monotone convergence theorem, the approximating occupation times converge to

Part I: Assume that

Part II: In general, we consider a Jordan chain,

Consider an eigen pair

Jordan chains: When some eigenvalues are not semi-simple, the proof follows the same idea, but the calculus becomes rather tedious. Therefore, we only present the main steps. Consider an arbitrary Jordan chain,

Analytic continuation: Finally, it remains to remove the assumption that the real part of every eigenvalue of

We now use analytic continuation in

Let us briefly return to the classical ruin concept, i.e., all

In order to obtain survival probabilities when

In the case of the classical Cramér–Lundberg model (

The simplicity of all the terms in Equation (

Moreover:

Let us finally consider a numerical illustration of our results for a Markov-modulated Cramér–Lundberg model Equation (

We use Theorem 4.2 to compute the vector of survival probabilities for zero initial capital:

Furthermore, Corollary 4.1 yields the vector of survival probabilities for an arbitrary initial capital,

Survival probabilities,

The probability of reaching level

Financial support by the Swiss National Science Foundation Project 200021-124635/1 is gratefully acknowledged.

The authors declare no conflict of interest.