A temporal approach to the Parisian risk model

2018 ◽  
Vol 55 (1) ◽  
pp. 302-317 ◽  
Author(s):  
Bin Li ◽  
Gordon E. Willmot ◽  
Jeff T. Y. Wong
Keyword(s):  
Risk Model ◽  
Levy Processes ◽  
New Approach ◽  
Approximation Approach ◽  
Parisian Ruin ◽  
Spatial Approximation ◽  
Underlying Processes ◽  
Spectrally Negative Lévy Processes ◽  
Ruin Problem ◽  
Observation Scheme

Abstract In this paper we propose a new approach to study the Parisian ruin problem for spectrally negative Lévy processes. Since our approach is based on a hybrid observation scheme switching between discrete and continuous observations, we call it a temporal approach as opposed to the spatial approximation approach in the literature. Our approach leads to a unified proof for the underlying processes with bounded or unbounded variation paths, and our result generalizes Loeffen et al. (2013).

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 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 Parisian ruin probability for spectrally negative Lévy processes

Bernoulli ◽  
2013 ◽  
Vol 19 (2) ◽  
pp. 599-609 ◽  
Author(s):  
Ronnie Loeffen ◽  
Irmina Czarna ◽  
Zbigniew Palmowski
Keyword(s):  
Ruin Probability ◽  
Lévy Processes ◽  
Levy Processes ◽  
Parisian Ruin ◽  
Spectrally Negative Lévy Processes

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

2014 ◽  
Vol 51 (04) ◽  
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 On the last exit times for spectrally negative Lévy processes

2017 ◽  
Vol 54 (2) ◽  
pp. 474-489 ◽  
Author(s):  
Yingqiu Li ◽  
Chuancun Yin ◽  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Exit Time ◽  
Levy Processes ◽  
Laplace Transforms ◽  
Occupation Time ◽  
Exit Times ◽  
New Approach ◽  
Last Exit Time ◽  
Spectrally Negative Lévy Processes

Abstract Using a new approach, for spectrally negative Lévy processes we find joint Laplace transforms involving the last exit time (from a semiinfinite interval), the value of the process at the last exit time, and the associated occupation time, which generalize some previous results.

scholarly journals A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 85 ◽  
Author(s):  
Mohamed Lkabous ◽  
Jean-François Renaud
Keyword(s):  
Risk Model ◽  
Risk Measures ◽  
Risk Measure ◽  
Short Paper ◽  
Classical Risk Model ◽  
Parisian Ruin ◽  
The Classical Risk Model

In this short paper, we study a VaR-type risk measure introduced by Guérin and Renaud and which is based on cumulative Parisian ruin. We derive some properties of this risk measure and we compare it to the risk measures of Trufin et al. and Loisel and Trufin.

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