scholarly journals Non-Parametric Threshold Estimation for the Wiener–Poisson Risk Model

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 506
Author(s):  
Honglong You ◽  
Yuan Gao
Keyword(s):  
Distribution Function ◽  
Survival Probability ◽  
Risk Model ◽  
Compound Poisson Process ◽  
Threshold Estimation ◽  
Threshold Method ◽  
Laplace Inversion ◽  
Jump Size ◽  
Consistency And Asymptotic Normality ◽  
Poisson Risk Model

In this paper, we consider the Wiener–Poisson risk model, which consists of a Wiener process and a compound Poisson process. Given the discrete record of observations, we use a threshold method and a regularized Laplace inversion technique to estimate the survival probability. In addition, we also construct an estimator for the distribution function of jump size and study its consistency and asymptotic normality. Finally, we give some simulations to verify our results.

scholarly journals Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Yan Li ◽  
Guoxin Liu
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Optimal Strategies ◽  
Proportional Reinsurance ◽  
Two Dimensional ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Optimal Value ◽  
Hamilton Jacobi Bellman ◽  
Poisson Risk Model

We consider the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. The optimization in the sense of minimizing the ruin probability which is defined by the sum of subportfolio is being ruined. Via the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal value function and prove the verification theorem. In addition, we obtain the Lundberg bounds and the Cramér-Lundberg approximation for the ruin probability and show that as the capital tends to infinity, the optimal strategies converge to the asymptotically optimal constant strategies. The asymptotic value can be found by maximizing the adjustment coefficient.

scholarly journals A multiple threshold method for fitting the generalized Pareto distribution and a simple representation of the rainfall process

2010 ◽  
Vol 7 (4) ◽  
pp. 4957-4994 ◽  
Author(s):  
R. Deidda
Keyword(s):  
Time Series ◽  
Distribution Function ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Daily Rainfall ◽  
Rainfall Time Series ◽  
Threshold Method ◽  
Generalized Pareto ◽  
Multiple Threshold ◽  
Optimum Threshold

Abstract. Previous studies indicate the generalized Pareto distribution (GPD) as a suitable distribution function to reliably describe the exceedances of daily rainfall records above a proper optimum threshold, which should be selected as small as possible to retain the largest sample while assuring an acceptable fitting. Such an optimum threshold may differ from site to site, affecting consequently not only the GPD scale parameter, but also the probability of threshold exceedance. Thus a first objective of this paper is to derive some expressions to parameterize a simple threshold-invariant three-parameter distribution function which is able to describe zero and non zero values of rainfall time series by assuring a perfect overlapping with the GPD fitted on the exceedances of any threshold larger than the optimum one. Since the proposed distribution does not depend on the local thresholds adopted for fitting the GPD, it will only reflect the on-site climatic signature and thus appears particularly suitable for hydrological applications and regional analyses. A second objective is to develop and test the Multiple Threshold Method (MTM) to infer the parameters of interest on the exceedances of a wide range of thresholds using again the concept of parameters threshold-invariance. We show the ability of the MTM in fitting historical daily rainfall time series recorded with different resolutions. Finally, we prove the supremacy of the MTM fit against the standard single threshold fit, often adopted for partial duration series, by evaluating and comparing the performances on Monte Carlo samples drawn by GPDs with different shape and scale parameters and different discretizations.

scholarly journals Dividend Moments in the Dual Risk Model: Exact and Approximate Approaches

2008 ◽  
Vol 38 (02) ◽  
pp. 399-422 ◽  
Author(s):  
Eric C.K. Cheung ◽  
Steve Drekic
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Size Distribution ◽  
Discrete Time ◽  
Continuous Time ◽  
Risk Model ◽  
Time Model ◽  
Jump Size ◽  
Time Of Ruin ◽  
The Time Of Ruin

In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in Avanzi et al. (2007). Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

scholarly journals Ruin Probabilities for Two Classes of Risk Processes

2005 ◽  
Vol 35 (1) ◽  
pp. 61-77 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Ruin Probability ◽  
Risk Model ◽  
Compound Poisson Process ◽  
Ruin Probabilities ◽  
Counting Processes ◽  
Arrival Times ◽  
The Laplace Transform ◽  
Sparre Andersen Risk Model ◽  
Risk Processes

We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

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