scholarly journals Property Claim Services by Compound Poisson Process And Inhomogeneous Levy Process

2018 ◽  
Vol 6 (1) ◽  
pp. 32
Author(s):  
Muhammed A. S. Murad
Keyword(s):  
Poisson Process ◽  
Lévy Process ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Arrival Process ◽  
Compound Poisson ◽  
Process Factor ◽  
Before And After ◽  
Exponential Lévy Process ◽  
Loss Index

In this paper, stochastic compound Poisson process is employed to value the catastrophic insurance options and model the claim arrival process for catastrophic events, which were written in the loss period , during which the catastrophe took place. Here, a time compound process gives the underlying loss index before and after  whose losses are revaluated by inhomogeneous exponential Levy process factor. For this paper, an exponential Levy process is used to evaluate the well-known European call option in order to price Property Claim Services catastrophe insurance based on catastrophe index.

scholarly journals Risk Theory with the Generalized Inverse Gaussian Lévy Process

2004 ◽  
Vol 34 (2) ◽  
pp. 361-377 ◽  
Author(s):  
Manuel Morales
Keyword(s):  
Lévy Process ◽  
Risk Model ◽  
Generalized Inverse ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Inverse Gaussian ◽  
Lévy Measure ◽  
Compound Poisson ◽  
Inverse Gaussian Process ◽  
Generalized Inverse Gaussian

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

scholarly journals Queues with Delays in Two-State Strategies and Lévy Input

2008 ◽  
Vol 45 (02) ◽  
pp. 314-332
Author(s):  
R. Bekker ◽  
O. J. Boxma ◽  
O. Kella
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
Poisson Process ◽  
Lévy Process ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Original Function ◽  
Steady State Distribution ◽  
State Distribution ◽  
Negative Drift

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

scholarly journals Queues with Delays in Two-State Strategies and Lévy Input

2008 ◽  
Vol 45 (2) ◽  
pp. 314-332 ◽  
Author(s):  
R. Bekker ◽  
O. J. Boxma ◽  
O. Kella
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
Poisson Process ◽  
Lévy Process ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Original Function ◽  
Steady State Distribution ◽  
State Distribution ◽  
Negative Drift

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

scholarly journals Continuous branching processes : the discrete hidden in the continuous : Dedicated to Claude Lobry

2008 ◽  
Vol Volume 9, 2007 Conference in... ◽  
Author(s):  
Etienne Pardoux
Keyword(s):  
Poisson Process ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Initial Condition ◽  
Compound Poisson ◽  
International Audience ◽  
Number Of Individuals ◽  
Branching Property

International audience Feller diffusion is a continuous branching process. The branching property tells us that for t > 0 fixed, when indexed by the initial condition, it is a subordinator (i. e. a positive–valued Lévy process), which is fact is a compound Poisson process. The number of points of this Poisson process can be interpreted as the number of individuals whose progeny survives during a number of generations of the order of t × N, where N denotes the size of the population, in the limit N ―>µ. This fact follows from recent results of Bertoin, Fontbona, Martinez [1]. We compare them with older results of de O’Connell [7] and [8]. We believe that this comparison is useful for better understanding these results. There is no new result in this presentation. La diffusion de Feller est un processus de branchement continu. La propriété de branchement nous dit que à t > 0 fixé, indexé par la condition initiale, ce processus est un subordinateur (processus de Lévy à valeurs positives), qui est en fait un processus de Poisson composé. Le nombre de points de ce processus de Poisson s’interprète comme le nombre d’individus dont la descendance survit au cours d’un nombre de générations de l’ordre de t × N, où N désigne la taille de la population, dans la limite N --> µ. Ce fait découle de résultats récents de Bertoin, Fontbona, Martinez [1]. Nous le rapprochons de résultats plus anciens de O’Connell [7] et [8]. Ce rapprochement nous semble aider à mieux comprendre ces résultats. Cet article ne contient pas de résultat nouveau.

scholarly journals Risk Theory with the Generalized Inverse Gaussian Lévy Process

2004 ◽  
Vol 34 (02) ◽  
pp. 361-377 ◽  
Author(s):  
Manuel Morales
Keyword(s):  
Lévy Process ◽  
Risk Model ◽  
Generalized Inverse ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Inverse Gaussian ◽  
Lévy Measure ◽  
Compound Poisson ◽  
Inverse Gaussian Process ◽  
Generalized Inverse Gaussian

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

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