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

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 $\omega(\cdot,\cdot)$ .

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

2005 ◽  
Vol 35 (02) ◽  
pp. 351-361 ◽  
Author(s):  
Andrew C.Y. Ng ◽  
Hailiang Yang
Keyword(s):  
Regime Switching ◽  
Joint Distribution ◽  
Risk Model ◽  
Insurance Risk ◽  
Additive Process ◽  
Deficit At Ruin ◽  
Markov Additive Process ◽  
The Deficit At Ruin ◽  
Markov Modulated ◽  
Markovian Regime Switching

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.

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

2005 ◽  
Vol 35 (2) ◽  
pp. 351-361
Author(s):  
Andrew C.Y. Ng ◽  
Hailiang Yang
Keyword(s):  
Regime Switching ◽  
Joint Distribution ◽  
Risk Model ◽  
Insurance Risk ◽  
Additive Process ◽  
Deficit At Ruin ◽  
Markov Additive Process ◽  
The Deficit At Ruin ◽  
Markov Modulated ◽  
Markovian Regime Switching

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.

scholarly journals Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

2016 ◽  
Vol 53 (2) ◽  
pp. 572-584 ◽  
Author(s):  
Erik J. Baurdoux ◽  
Juan Carlos Pardo ◽  
José Luis Pérez ◽  
Jean-François Renaud
Keyword(s):  
Risk Process ◽  
Levy Processes ◽  
Insurance Company ◽  
Excursion Theory ◽  
Insurance Risk ◽  
Scale Functions ◽  
Surplus Process ◽  
Parisian Ruin ◽  
Risk Processes ◽  
Spectrally Negative Lévy Processes

Abstract Inspired by the works of Landriault et al. (2011), (2014), we study the Gerber–Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of so-called scale functions. In particular, we extend the recent results of Landriault et al. (2011), (2014).

scholarly journals A sample path approach to mean busy periods for Markov-modulated queues and fluids

1994 ◽  
Vol 26 (4) ◽  
pp. 1117-1121 ◽  
Author(s):  
Søren Asmussen ◽  
Mogens Bladt
Keyword(s):  
Time Reversal ◽  
Busy Period ◽  
Fluid Model ◽  
Sample Path ◽  
Steady State Distribution ◽  
State Distribution ◽  
Additive Process ◽  
Markov Additive Process ◽  
The Mean ◽  
Markov Modulated

The mean busy period of a Markov-modulated queue or fluid model is computed by an extension of the time-reversal argument connecting the steady-state distribution and the maximum of a related Markov additive process.

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

2010 ◽  
Vol 47 (2) ◽  
pp. 441-458 ◽  
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.

scholarly journals A sample path approach to mean busy periods for Markov-modulated queues and fluids

1994 ◽  
Vol 26 (04) ◽  
pp. 1117-1121 ◽  
Author(s):  
Søren Asmussen ◽  
Mogens Bladt
Keyword(s):  
Time Reversal ◽  
Busy Period ◽  
Fluid Model ◽  
Sample Path ◽  
Steady State Distribution ◽  
State Distribution ◽  
Additive Process ◽  
Markov Additive Process ◽  
The Mean ◽  
Markov Modulated

The mean busy period of a Markov-modulated queue or fluid model is computed by an extension of the time-reversal argument connecting the steady-state distribution and the maximum of a related Markov additive process.

Draw-down Parisian ruin for spectrally negative Lévy processes

2020 ◽  
Vol 52 (4) ◽  
pp. 1164-1196
Author(s):  
Wenyuan Wang ◽  
Xiaowen Zhou
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Fixed Amount ◽  
Ruin Time ◽  
Surplus Process ◽  
Potential Measure ◽  
Parisian Ruin ◽  
Exit Problems ◽  
Risk Processes ◽  
Spectrally Negative Lévy Processes

AbstractDraw-down time for a stochastic process is the first passage time of a draw-down level that depends on the previous maximum of the process. In this paper we study the draw-down-related Parisian ruin problem for spectrally negative Lévy risk processes. Intuitively, a draw-down Parisian ruin occurs when the surplus process has continuously stayed below the dynamic draw-down level for a fixed amount of time. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit problems via excursion theory. We also find an expression for the potential measure for the process killed at the draw-down Parisian time. As applications, we obtain new results for spectrally negative Lévy risk processes with dividend barrier and with Parisian ruin.

Markov-modulated linear fluid networks with Markov additive input

2002 ◽  
Vol 39 (02) ◽  
pp. 413-420 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Markov Chain ◽  
External Input ◽  
Release Rates ◽  
Constant Multiple ◽  
Additive Process ◽  
Markov Additive Process ◽  
Fluid Networks ◽  
Markov Modulated

We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.

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

2010 ◽  
Vol 47 (02) ◽  
pp. 441-458 ◽  
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.

Markov-modulated linear fluid networks with Markov additive input

2002 ◽  
Vol 39 (2) ◽  
pp. 413-420 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Markov Chain ◽  
External Input ◽  
Release Rates ◽  
Constant Multiple ◽  
Additive Process ◽  
Markov Additive Process ◽  
Fluid Networks ◽  
Markov Modulated

We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.

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