Lundberg-type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin under a Markov-modulated Risk Model

In this paper, we consider a Markov-modulated risk model (also called Markovian regime switching insurance risk model). Follow Asmussen (2000, 2003), by using the theory of Markov additive process, an exponential martingale is constructed and Lundberg-type upper bounds for the joint distribution of surplus immediately before and at ruin are obtained. As a natural corollary, bounds for the distribution of the deficit at ruin are obtained. We also present some numerical results to illustrate the tightness of the bound obtained in this paper.

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Fluctuation identities for Omega-killed spectrally negative Markov additive processes and dividend problem

Advances in Applied Probability ◽

10.1017/apr.2020.2 ◽

2020 ◽

Vol 52 (2) ◽

pp. 404-432

Author(s):

Irmina Czarna ◽

Adam Kaszubowski ◽

Shu Li ◽

Zbigniew Palmowski

Keyword(s):

Insurance Risk ◽

Additive Process ◽

Markov Additive Process ◽

Surplus Process ◽

Current State ◽

Markov Modulated ◽

Exit Problems ◽

State Of The Environment ◽

Risk Processes ◽

Omega Model

AbstractIn this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of new generalizations of classical scale matrices for MAPs. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the Omega model, where bankruptcy takes place at rate $\omega(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider Markov-modulated Brownian motion (MMBM) as a special case and present analytical and numerical results for a particular choice of piecewise intensity function $\omega(\cdot,\cdot)$ .

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A quintuple law for Markov additive processes with phase-type jumps

Journal of Applied Probability ◽

10.1239/jap/1276784902 ◽

2010 ◽

Vol 47 (2) ◽

pp. 441-458 ◽

Cited By ~ 17

Author(s):

Lothar Breuer

Keyword(s):

Joint Distribution ◽

First Passage Times ◽

Insurance Risk ◽

Maximal Level ◽

First Passage ◽

Additive Process ◽

Process Map ◽

Markov Additive Process ◽

Phase Type ◽

Passage Times

We consider a Markov additive process (MAP) with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. A marginal of the derived distribution is the Gerber-Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.

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A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

Journal of Applied Probability ◽

10.1017/jpr.2016.99 ◽

2017 ◽

Vol 54 (1) ◽

pp. 267-285 ◽

Cited By ~ 2

Author(s):

Onno J. Boxma ◽

Esther Frostig ◽

David Perry

Keyword(s):

Risk Model ◽

Risk Process ◽

Insurance Risk ◽

Deficit At Ruin ◽

Scale Functions ◽

The Deficit At Ruin ◽

The Laplace Transform ◽

Surplus Before Ruin ◽

Set Up ◽

Spectrally Negative Lévy Processes

AbstractWe consider a Cramér–Lundberg insurance risk process with the added feature of reinsurance. If an arriving claim finds the reserve below a certain threshold γ, or if it would bring the reserve below that level, then a reinsurer pays part of the claim. Using fluctuation theory and the theory of scale functions of spectrally negative Lévy processes, we derive expressions for the Laplace transform of the time to ruin and of the joint distribution of the deficit at ruin and the surplus before ruin. We specify these results in much more detail for the threshold set-up in the case of proportional reinsurance.

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A quintuple law for Markov additive processes with phase-type jumps

Journal of Applied Probability ◽

10.1017/s0021900200006744 ◽

2010 ◽

Vol 47 (02) ◽

pp. 441-458 ◽

Cited By ~ 4

Author(s):

Lothar Breuer

Keyword(s):

Joint Distribution ◽

First Passage Times ◽

Insurance Risk ◽

Maximal Level ◽

First Passage ◽

Additive Process ◽

Process Map ◽

Markov Additive Process ◽

Phase Type ◽

Passage Times

We consider a Markov additive process (MAP) with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. A marginal of the derived distribution is the Gerber-Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.

Download Full-text