Gerber–Shiu Function in a Class of Delayed and Perturbed Risk Model with Dependence

This paper considers the risk model perturbed by a diffusion process with a time delay in the arrival of the first two claims and takes into account dependence between claim amounts and the claim inter-occurrence times. Assuming that the time arrival of the first claim follows a generalized mixed equilibrium distribution, we derive the integro-differential Equations of the Gerber–Shiu function and its defective renewal equations. For the situation where claim amounts follow exponential distribution, we provide an explicit expression of the Gerber–Shiu function. Numerical examples are provided to illustrate the ruin probability.

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Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

Mathematics ◽

10.3390/math9090982 ◽

2021 ◽

Vol 9 (9) ◽

pp. 982

Author(s):

Yujuan Huang ◽

Jing Li ◽

Hengyu Liu ◽

Wenguang Yu

Keyword(s):

Fourier Series ◽

Ruin Probability ◽

Risk Model ◽

Expansion Method ◽

Nonparametric Estimator ◽

Insurance Risk ◽

Numerical Examples ◽

Complex Fourier Series ◽

Premium Income ◽

Insurance Risk Model

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

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The Perturbed Dual Risk Model with Constant Interest and a Threshold Dividend Strategy

Abstract and Applied Analysis ◽

10.1155/2013/981076 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Fanzi Zeng ◽

Jisheng Xu

Keyword(s):

Ruin Probability ◽

Risk Model ◽

Present Value ◽

Numerical Examples ◽

Threshold Dividend Strategy ◽

Absolute Ruin ◽

Dual Risk Model ◽

The Moment ◽

The Impact ◽

Constant Interest

We consider the perturbed dual risk model with constant interest and a threshold dividend strategy. Firstly, we investigate the moment-generation function of the present value of total dividends until ruin. Integrodifferential equations with certain boundary conditions are derived for the present value of total dividends. Furthermore, using techniques of sinc numerical methods, we obtain the approximation results to the expected present value of total dividends. Finally, numerical examples are presented to show the impact of interest on the expected present value of total dividends and the absolute ruin probability.

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Ruin Probabilities of a Discrete-time Multi-risk Model

Information Technology And Control ◽

10.5755/j01.itc.44.4.8635 ◽

2015 ◽

Vol 44 (4) ◽

pp. 367-379 ◽

Cited By ~ 2

Author(s):

Andrius Grigutis ◽

Agneška Korvel ◽

Jonas Šiaulys

Keyword(s):

Discrete Time ◽

Ruin Probability ◽

Arrival Time ◽

Risk Model ◽

Recursive Formula ◽

Ruin Probabilities ◽

Numerical Examples ◽

Premium Rate ◽

Insurance Business ◽

Independent Series

In this work, we investigate a multi-risk model describing insurance business with two or more independent series of claim amounts. Each series of claim amounts consists of independent nonnegative random variables. Claims of each series occur periodically with some fixed inter-arrival time. Claim amounts occur until they can be compensated by a common premium rate and the initial insurer's surplus. In this article, wederive a recursive formula for calculation of finite-time ruin probabilities. In the case of bi-risk model, we present a procedure to calculate the ultimate ruin probability. We add several numerical examples illustrating application of the derived formulas.DOI: http://dx.doi.org/10.5755/j01.itc.44.4.8635

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Studies on a Double Poisson-Geometric Insurance Risk Model with Interference

Discrete Dynamics in Nature and Society ◽

10.1155/2013/128796 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Yujuan Huang ◽

Wenguang Yu

Keyword(s):

Ruin Probability ◽

Laplace Transformation ◽

Risk Model ◽

Stopping Time ◽

Adjustment Coefficient ◽

Insurance Risk ◽

Numerical Examples ◽

Lundberg Inequality ◽

First Time ◽

Insurance Risk Model

This paper mainly studies a generalized double Poisson-Geometric insurance risk model. By martingale and stopping time approach, we obtain adjustment coefficient equation, the Lundberg inequality, and the formula for the ruin probability. Also the Laplace transformation of the time when the surplus reaches a given level for the first time is discussed, and the expectation and its variance are obtained. Finally, we give the numerical examples.

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On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance

Journal of Applied Mathematics ◽

10.1155/2018/9180780 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Christian Kasumo ◽

Juma Kasozi ◽

Dmitry Kuznetsov

Keyword(s):

Ruin Probability ◽

Risk Model ◽

Insurance Company ◽

Block Method ◽

Numerical Examples ◽

Volterra Integrodifferential Equation ◽

Heavy Tailed Distributions ◽

Heavy Tailed ◽

Hamilton Jacobi Bellman ◽

And Diffusion

We consider an insurance company whose reserves dynamics follow a diffusion-perturbed risk model. To reduce its risk, the company chooses to reinsure using proportional or excess-of-loss reinsurance. Using the Hamilton-Jacobi-Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation (VIDE) which we transform into a linear Volterra integral equation (VIE) of the second kind. We then proceed to solve this linear VIE numerically using the block-by-block method for the optimal reinsurance policy that minimizes the ultimate ruin probability for the chosen parameters. Numerical examples with both light- and heavy-tailed distributions are given. The results show that proportional reinsurance increases the survival of the company in both light- and heavy-tailed distributions for the Cramér-Lundberg and diffusion-perturbed models.

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Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy

Mathematics ◽

10.3390/math8111885 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1885

Author(s):

Olena Ragulina ◽

Jonas Šiaulys

Keyword(s):

Ruin Probability ◽

Risk Model ◽

Upper Bounds ◽

Explicit Formulas ◽

Exponential Bound ◽

Numerical Examples ◽

Exponential Bounds ◽

Erlang Distributions

This paper is devoted to the investigation of the ruin probability in the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. We obtain an exponential bound for the ruin probability and investigate conditions, under which it holds for a number of distributions of the premium and claim sizes. Next, we use the exponential bound to construct non-exponential bounds for the ruin probability. We show that the non-exponential bounds turn out to be tighter than the exponential one in some cases. Moreover, we derive explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions and apply them to investigate how tight the bounds are. To illustrate and analyze the results obtained, we give numerical examples.

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On the Parisian ruin of the dual Lévy risk model

Journal of Applied Probability ◽

10.1017/jpr.2017.59 ◽

2017 ◽

Vol 54 (4) ◽

pp. 1193-1212 ◽

Cited By ~ 1

Author(s):

Chen Yang ◽

Kristian P. Sendova ◽

Zhong Li

Keyword(s):

Ruin Probability ◽

Risk Model ◽

Drift Diffusion Model ◽

Drift Diffusion ◽

Expected Discounted Penalty Function ◽

Numerical Examples ◽

Discounted Penalty Function ◽

Usual Concept ◽

Parisian Ruin ◽

Expected Discounted Penalty

AbstractIn this paper we investigate the Parisian ruin problem of the general dual Lévy risk model. Unlike the usual concept of ultimate ruin, allowing the surplus level to be negative within a prespecified period indicates that the deficit at Parisian ruin is not necessarily equal to zero. Hence, we consider a Gerber–Shiu type expected discounted penalty function at the Parisian ruin and obtain an explicit expression for this function under the dual Lévy risk model. As particular cases, we calculate the Parisian ruin probability and the expected discountedkth moments of the deficit at the Parisian ruin for the compound Poisson dual risk model and a drift-diffusion model. Numerical examples are given to illustrate the behavior of Parisian ruin and the expected discounted deficit at Parisian ruin.

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RECURSIVE CALCULATION OF RUIN PROBABILITIES AT OR BEFORE CLAIM INSTANTS FOR NON-IDENTICALLY DISTRIBUTED CLAIMS

Astin Bulletin ◽

10.1017/asb.2014.30 ◽

2014 ◽

Vol 45 (2) ◽

pp. 421-443 ◽

Cited By ~ 13

Author(s):

Anisoara Maria Raducan ◽

Raluca Vernic ◽

Gheorghita Zbaganu

Keyword(s):

Continuous Time ◽

Ruin Probability ◽

Risk Model ◽

Recursive Algorithm ◽

Risk Process ◽

Ruin Probabilities ◽

Numerical Examples ◽

Claim Number ◽

Recursive Calculation ◽

The Impact

AbstractIn this paper, we present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model. The claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times (ICTs). The claim sizes (CSs) are independent and distributed in Erlang's family, i.e., they can have different parameters, which yields a non-homogeneous risk process. We present the corresponding recursive algorithm used to evaluate the above mentioned ruin probability and we illustrate it on several numerical examples in which we vary the model's parameters to assess the impact of the non-homogeneity on the resulting ruin probability.

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