Gerber–Shiu Function at Draw-Down Parisian Ruin Time for the Spectrally Negative Lévy Risk Process

2021 ◽  
Author(s):  
Aili Zhang
Keyword(s):  
Risk Process ◽  
Ruin Time ◽  
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).

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.

scholarly journals A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Aili Zhang ◽  
Zhang Liu
Keyword(s):  
Risk Model ◽  
Poisson Model ◽  
Risk Process ◽  
Closed Form Expression ◽  
Form Expression ◽  
Ruin Time ◽  
Scale Functions ◽  
Surplus Process ◽  
Dividend Payments ◽  
Special Cases

This paper focuses on the De Finetti’s dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

scholarly journals An Uncertain Alternating Renewal Insurance Risk Model

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jia Zhai ◽  
Haitao Zheng ◽  
Manying Bai ◽  
Yunyun Jiang
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Insurance Risk ◽  
Uncertain Measure ◽  
Uncertainty Distribution ◽  
Ruin Time ◽  
Reward Process ◽  
Renewal Reward Process ◽  
Explicit Expressions ◽  
Insurance Risk Model

The claim process in an insurance risk model with uncertainty is traditionally described by an uncertain renewal reward process. However, the claim process actually includes two processes, which are called the report process and the payment process, respectively. An alternative way is to describe the claim process by an uncertain alternating renewal reward process. Therefore, this paper proposes an insurance risk model under uncertain measure in which the claim process is supposed to be an alternating renewal reward process and the premium process is regarded as a renewal reward process. Then, the paper also gives the inverse uncertainty distribution of the insurance risk process. The expression of ruin index and the uncertainty distribution of the ruin time are derived which both have explicit expressions based on given uncertainty distributions. Finally, several examples are provided to illustrate the modeling ideas.

scholarly journals On the Time Spent in the Red by a Refracted Lévy Risk Process

2014 ◽  
Vol 51 (4) ◽  
pp. 1171-1188 ◽  
Author(s):  
Jean-François Renaud
Keyword(s):  
Lévy Processes ◽  
Risk Process ◽  
Levy Processes ◽  
Occupation Times ◽  
Premium Rate ◽  
Probability Of Bankruptcy ◽  
Parisian Ruin ◽  
Spectrally Negative Lévy Processes

In this paper we introduce an insurance ruin model with an adaptive premium rate, henceforth referred to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the wealth process recovers. The analysis is focused mainly on the time a refracted Lévy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from [11] and [16], we identify the distribution of various functionals related to occupation times of refracted spectrally negative Lévy processes. For example, these results are used to compute both the probability of bankruptcy and the probability of Parisian ruin in this model with restructuring.

scholarly journals An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density

2009 ◽  
Vol 46 (01) ◽  
pp. 85-98 ◽  
Author(s):  
R. L. Loeffen
Keyword(s):  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Lévy Measure ◽  
Ruin Time ◽  
Optimal Dividends ◽  
Spectrally Negative Lévy Process ◽  
Completely Monotone ◽  
Monotone Density ◽  
Spectrally Negative Lévy Processes

We consider a modified version of the classical optimal dividends problem of de Finetti in which the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Lévy process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Lévy measure has a completely monotone density. As a prerequisite for the proof, we show that, under the aforementioned condition on the Lévy measure, theq-scale function of the spectrally negative Lévy process has a derivative which is strictly log-convex.

scholarly journals An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density

2009 ◽  
Vol 46 (1) ◽  
pp. 85-98 ◽  
Author(s):  
R. L. Loeffen
Keyword(s):  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Lévy Measure ◽  
Ruin Time ◽  
Optimal Dividends ◽  
Spectrally Negative Lévy Process ◽  
Completely Monotone ◽  
Monotone Density ◽  
Spectrally Negative Lévy Processes

We consider a modified version of the classical optimal dividends problem of de Finetti in which the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Lévy process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Lévy measure has a completely monotone density. As a prerequisite for the proof, we show that, under the aforementioned condition on the Lévy measure, the q-scale function of the spectrally negative Lévy process has a derivative which is strictly log-convex.

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.

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