scholarly journals Ruin Probability in Compound Poisson Process with Investment

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Yong Wu ◽  
Xiang Hu
Keyword(s):  
Differential Equations ◽  
Poisson Process ◽  
Ruin Probability ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Compound Poisson ◽  
Risky Assets ◽  
Black Scholes Model ◽  
Black Scholes

We consider that the surplus of an insurer follows compound Poisson process and the insurer would invest its surplus in risky assets, whose prices satisfy the Black-Scholes model. In the risk process, we decompose the ruin probability into the sum of two ruin probabilities which are caused by the claim and the oscillation, respectively. We derive the integro-differential equations for these ruin probabilities these ruin probabilities. When the claim sizes are exponentially distributed, third-order differential equations of the ruin probabilities are derived from the integro-differential equations and a lower bound is obtained.

scholarly journals Hubungan Antara Brownian Motion (The Winner Process) dan Surplus Process

2012 ◽  
Vol 12 (1) ◽  
pp. 47
Author(s):  
Tohap Manurung
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Wiener Process ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Actuarial Science ◽  
Compound Poisson ◽  
Brownian Motion With Drift ◽  
Surplus Process ◽  
The Standard Model

HUBUNGAN ANTARA BROWNIAN MOTION (THE WIENER PROCESS) DAN SURPLUS PROCESS ABSTRAK Suatu analisis model continous-time menjadi cakupan yang akan dibahas dalam tulisan ini. Dengan demikian pengenalan proses stochastic akan sangat berperan. Dua proses akan di analisis yaitu proses compound Poisson dan Brownian motion. Proses compound Poisson sudah menjadi model standard untuk Ruin analysis dalam ilmu aktuaria. Sementara Brownian motion sangat berguna dalam teori keuangan modern dan juga dapat digunakan sebagai approksimasi untuk proses compound Poisson. Hal penting dalam tulisan ini adalah menujukkan bagaimana surplus process berdasarkan proses resiko compound Poisson dihubungkan dengan Brownian motion with Drift Process. Kata kunci: Brownian motion with Drift process, proses surplus, compound Poisson   RELATIONSHIP  BETWEEN  BROWNIAN MOTION (THE WIENER PROCESS) AND THE SURPLUS PROCESS ABSTRACT An analysis of continous-time models is covered in this paper. Thus, this requires an introduction to stochastic processes. Two processes are analyzed: the compound Poisson process and Brownian motion. The compound Poisson process has been the standard model for ruin analysis in actuarial science, while Brownian motion has found considerable use in modern financial theory and also can be used as an approximation to the compound Pisson process. The important thing is to show how the surplus process based on compound poisson risk process is related to Brownian motion with drift process. Keywords: Brownian motion with drift process, surplus process, compound Poisson

scholarly journals Large deviations for risk processes with reinsurance

2006 ◽  
Vol 43 (03) ◽  
pp. 713-728 ◽  
Author(s):  
Claudio Macci ◽  
Gabriele Stabile
Keyword(s):  
Large Deviations ◽  
Poisson Process ◽  
Large Deviation ◽  
Compound Poisson Process ◽  
Ruin Probabilities ◽  
Small Claim ◽  
Compound Poisson ◽  
Numerical Example ◽  
Markov Modulated ◽  
Risk Processes

We consider risk processes with reinsurance. A general family of reinsurance contracts is allowed, including proportional and excess-of-loss policies. Claim occurrence is regulated by a classical compound Poisson process or by a Markov-modulated compound Poisson process. We provide some large deviation results concerning these two risk processes in the small-claim case. Finally, we derive the so-called Lundberg estimate for the ruin probabilities and present a numerical example.

scholarly journals Ruin in the perturbed compound Poisson risk process under interest force

2005 ◽  
Vol 37 (03) ◽  
pp. 819-835 ◽  
Author(s):  
Jun Cai ◽  
Hailiang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Process ◽  
Integrodifferential Equations ◽  
Ruin Probabilities ◽  
Compound Poisson ◽  
Interest Force ◽  
Stochastic Interest ◽  
Constant Interest Force ◽  
Constant Interest

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

scholarly journals Ruin in the perturbed compound Poisson risk process under interest force

2005 ◽  
Vol 37 (3) ◽  
pp. 819-835 ◽  
Author(s):  
Jun Cai ◽  
Hailiang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Process ◽  
Integrodifferential Equations ◽  
Ruin Probabilities ◽  
Compound Poisson ◽  
Interest Force ◽  
Stochastic Interest ◽  
Constant Interest Force ◽  
Constant Interest

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

scholarly journals On the Numerical Evaluation of Stop-Loss Premiums

1979 ◽  
Vol 10 (3) ◽  
pp. 318-324 ◽  
Author(s):  
F. Covens ◽  
M. Van Wouwe ◽  
M. Goovaerts
Keyword(s):  
Poisson Process ◽  
Numerical Evaluation ◽  
Fourier Transforms ◽  
Numerical Procedure ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Compound Poisson ◽  
Stop Loss

A numerical procedure is described to evaluate the stop-loss premium in case the risk process is a compound Poisson process. The method is mainly based on an algorithm of R. Piessens and M. Branders for the numerical evaluation of Fourier transforms.

scholarly journals Large deviations for risk processes with reinsurance

2006 ◽  
Vol 43 (3) ◽  
pp. 713-728 ◽  
Author(s):  
Claudio Macci ◽  
Gabriele Stabile
Keyword(s):  
Large Deviations ◽  
Poisson Process ◽  
Large Deviation ◽  
Compound Poisson Process ◽  
Ruin Probabilities ◽  
Small Claim ◽  
Compound Poisson ◽  
Numerical Example ◽  
Markov Modulated ◽  
Risk Processes

We consider risk processes with reinsurance. A general family of reinsurance contracts is allowed, including proportional and excess-of-loss policies. Claim occurrence is regulated by a classical compound Poisson process or by a Markov-modulated compound Poisson process. We provide some large deviation results concerning these two risk processes in the small-claim case. Finally, we derive the so-called Lundberg estimate for the ruin probabilities and present a numerical example.

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