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.

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).

General drawdown-based de Finetti optimization for spectrally negative Lévy risk processes

2018 ◽  
Vol 55 (2) ◽  
pp. 513-542 ◽  
Author(s):  
Wenyuan Wang ◽  
Xiaowen Zhou
Keyword(s):  
Value Function ◽  
Small Class ◽  
General Version ◽  
Ruin Time ◽  
Underlying Process ◽  
Optimal Dividend ◽  
Running Maximum ◽  
Admissible Strategies ◽  
Risk Processes ◽  
Spectrally Negative Lévy Processes

Abstract For spectrally negative Lévy risk processes we consider a general version of de Finetti's optimal dividend problem in which the ruin time is replaced with a general drawdown time from the running maximum in its value function. We identify a condition under which a barrier dividend strategy is optimal among all admissible strategies if the underlying process does not belong to a small class of compound Poisson processes with drift, for which the take-the-money-and-run dividend strategy is optimal. It generalizes the previous results on dividend optimization from ruin time based to drawdown time based. The associated drawdown functions are discussed in detail for examples of spectrally negative Lévy processes.

scholarly journals Parisian ruin of self-similar Gaussian risk processes

2015 ◽  
Vol 52 (3) ◽  
pp. 688-702 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Enkelejd Hashorva ◽  
Lanpeng Ji
Keyword(s):  
Normal Approximation ◽  
Asymptotic Relation ◽  
Exact Asymptotics ◽  
Ruin Time ◽  
Self Similar ◽  
Parisian Ruin ◽  
Risk Processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

Spectrally negative Lévy processes with applications in risk theory

2001 ◽  
Vol 33 (1) ◽  
pp. 281-291 ◽  
Author(s):  
Hailiang Yang ◽  
Lianzeng Zhang
Keyword(s):  
Lévy Processes ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Levy Processes ◽  
Point Of View ◽  
Ruin Time ◽  
Classical Risk Process ◽  
Risk Processes ◽  
First Time ◽  
Spectrally Negative Lévy Processes

In this paper, results on spectrally negative Lévy processes are used to study the ruin probability under some risk processes. These processes include the compound Poisson process and the gamma process, both perturbed by diffusion. In addition, the first time the risk process hits a given level is also studied. In the case of classical risk process, the joint distribution of the ruin time and the first recovery time is obtained. Some results in this paper have appeared before (e.g., Dufresne and Gerber (1991), Gerber (1990), dos Reis (1993)). We revisit them from the Lévy process theory's point of view and in a unified and simple way.

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 Ruin Probability with Parisian Delay for a Spectrally Negative Lévy Risk Process

2011 ◽  
Vol 48 (4) ◽  
pp. 984-1002 ◽  
Author(s):  
Irmina Czarna ◽  
Zbigniew Palmowski
Keyword(s):  
Ruin Probability ◽  
Risk Process ◽  
Insurance Risk ◽  
Fixed Amount ◽  
Surplus Process ◽  
Parisian Ruin

In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.

scholarly journals Ruin Probability with Parisian Delay for a Spectrally Negative Lévy Risk Process

2011 ◽  
Vol 48 (04) ◽  
pp. 984-1002 ◽  
Author(s):  
Irmina Czarna ◽  
Zbigniew Palmowski
Keyword(s):  
Ruin Probability ◽  
Risk Process ◽  
Insurance Risk ◽  
Fixed Amount ◽  
Surplus Process ◽  
Parisian Ruin

In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.

scholarly journals Exit problems for general draw-down times of spectrally negative Lévy processes

2019 ◽  
Vol 56 (2) ◽  
pp. 441-457 ◽  
Author(s):  
Bo Li ◽  
Nhat Linh Vu ◽  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Exit Time ◽  
Levy Processes ◽  
Laplace Transforms ◽  
Scale Functions ◽  
Dynamic Level ◽  
Running Maximum ◽  
Potential Measure ◽  
Exit Problems ◽  
Spectrally Negative Lévy Processes

AbstractFor spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

scholarly journals Exit Problems for Jump Processes Having Double-Sided Jumps with Rational Laplace Transforms

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yuzhen Wen ◽  
Chuancun Yin
Keyword(s):  
Laplace Transform ◽  
Joint Distribution ◽  
First Passage Time ◽  
Jump Process ◽  
Passage Time ◽  
Jump Processes ◽  
Threshold Strategy ◽  
First Passage ◽  
Exit Problem ◽  
Exit Problems

We consider the two-sided first-exit problem for a jump process having jumps with rational Laplace transform. We derive the joint distribution of the first passage time to two-sided barriers and the value of process at the first passage time. As applications, we present explicit expressions of the dividend formulae for barrier strategy and threshold strategy.

scholarly journals Parisian ruin of self-similar Gaussian risk processes

2015 ◽  
Vol 52 (03) ◽  
pp. 688-702 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Enkelejd Hashorva ◽  
Lanpeng Ji
Keyword(s):  
Normal Approximation ◽  
Asymptotic Relation ◽  
Exact Asymptotics ◽  
Ruin Time ◽  
Self Similar ◽  
Parisian Ruin ◽  
Risk Processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

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