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Journal articles on the topic 'The Craér-Lundberg risk model'

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Published: 30 May 2022

Last updated: 31 May 2022

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1

Albrecher, Hansjörg, Sem C. Borst, Onno J. Boxma, and Jacques Resing. "Ruin excursions, the G/G/∞ queue, and tax payments in renewal risk models." Journal of Applied Probability 48, A (August 2011): 3–14. http://dx.doi.org/10.1017/s0021900200099083.

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In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramér–Lundberg model with and without tax payments. We also provide a relation of the Cramér–Lundberg risk model with the G/G/∞ queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cramér– Lundberg risk model.

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2

Albrecher, Hansjörg, Sem C. Borst, Onno J. Boxma, and Jacques Resing. "Ruin excursions, the G/G/∞ queue, and tax payments in renewal risk models." Journal of Applied Probability 48, A (August 2011): 3–14. http://dx.doi.org/10.1239/jap/1318940451.

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In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramér–Lundberg model with and without tax payments. We also provide a relation of the Cramér–Lundberg risk model with the G/G/∞ queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cramér– Lundberg risk model.

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3

Yuen, Kam C., Junyi Guo, and Kai W. Ng. "On Ultimate Ruin in a Delayed-Claims Risk Model." Journal of Applied Probability 42, no. 01 (March 2005): 163–74. http://dx.doi.org/10.1017/s0021900200000139.

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In this paper, we consider a risk model in which each main claim induces a delayed claim called a by-claim. The time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. From martingale theory, an expression for the ultimate ruin probability can be derived using the Lundberg exponent of the associated nondelayed risk model. It can be shown that the Lundberg exponent of the proposed risk model is the same as that of the nondelayed one. Brownian motion approximations for ruin probabilities are also discussed.

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4

Yuen, Kam C., Junyi Guo, and Kai W. Ng. "On Ultimate Ruin in a Delayed-Claims Risk Model." Journal of Applied Probability 42, no. 1 (March 2005): 163–74. http://dx.doi.org/10.1239/jap/1110381378.

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In this paper, we consider a risk model in which each main claim induces a delayed claim called a by-claim. The time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. From martingale theory, an expression for the ultimate ruin probability can be derived using the Lundberg exponent of the associated nondelayed risk model. It can be shown that the Lundberg exponent of the proposed risk model is the same as that of the nondelayed one. Brownian motion approximations for ruin probabilities are also discussed.

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5

Schmidli, Hanspeter. "Lundberg inequalities for a Cox model with a piecewise constant intensity." Journal of Applied Probability 33, no. 1 (March 1996): 196–210. http://dx.doi.org/10.2307/3215277.

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A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

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6

Schmidli, Hanspeter. "Lundberg inequalities for a Cox model with a piecewise constant intensity." Journal of Applied Probability 33, no. 01 (March 1996): 196–210. http://dx.doi.org/10.1017/s0021900200103857.

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A Cox risk process with a piecewise constant intensity is considered where the sequence (Li ) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li . In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

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7

Minkova, Leda D. "The Pólya-Aeppli process and ruin problems." Journal of Applied Mathematics and Stochastic Analysis 2004, no. 3 (January 1, 2004): 221–34. http://dx.doi.org/10.1155/s1048953304309032.

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The Pólya-Aeppli process as a generalization of the homogeneous Poisson process is defined. We consider the risk model in which the counting process is the Pólya-Aeppli process. It is called a Pólya-Aeppli risk model. The problem of finding the ruin probability and the Cramér-Lundberg approximation is studied. The Cramér condition and the Lundberg exponent are defined. Finally, the comparison between the Pélya-Aeppli risk model and the corresponding classical risk model is given.

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8

Rodríguez-Martínez, Eugenio V., Rui M. R. Cardoso, and Alfredo D. Egídio dos Reis. "SOME ADVANCES ON THE ERLANG(n) DUAL RISK MODEL." ASTIN Bulletin 45, no. 1 (August 27, 2014): 127–50. http://dx.doi.org/10.1017/asb.2014.19.

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AbstractThe dual risk model assumes that the surplus of a company decreases at a constant rate over time and grows by means of upward jumps, which occur at random times and sizes. It is said to have applications to companies with economical activities involved in research and development. This model is dual to the well-known Cramér-Lundberg risk model with applications to insurance. Most existing results on the study of the dual model assume that the random waiting times between consecutive gains follow an exponential distribution, as in the classical Cramér-Lundberg risk model. We generalize to other compound renewal risk models where such waiting times are Erlang(n) distributed. Using the roots of the fundamental and the generalized Lundberg's equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Furthermore, we compute expected discounted dividends, as well as higher moments, when the individual common gains follow a Phase-Type, PH(m), distribution. We also perform illustrations working some examples for some particular gain distributions and obtain numerical results.

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9

Cossette, Hélène, Etienne Marceau, and Véronique Maume-Deschamps. "Discrete-Time Risk Models Based on Time Series for Count Random Variables." ASTIN Bulletin 40, no. 1 (May 2010): 123–50. http://dx.doi.org/10.2143/ast.40.1.2049221.

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AbstractIn this paper, we consider various specifications of the general discrete-time risk model in which a serial dependence structure is introduced between the claim numbers for each period. We consider risk models based on compound distributions assuming several examples of discrete variate time series as specific temporal dependence structures: Poisson MA(1) process, Poisson AR(1) process, Markov Bernoulli process and Markov regime-switching process. In these models, we derive expressions for a function that allow us to find the Lundberg coefficient. Specific cases for which an explicit expression can be found for the Lundberg coefficient are also presented. Numerical examples are provided to illustrate different topics discussed in the paper.

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10

Li, Yan, and Guoxin Liu. "Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model." Discrete Dynamics in Nature and Society 2012 (2012): 1–26. http://dx.doi.org/10.1155/2012/802518.

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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.

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11

Andrulytė, Ieva Marija, Emilija Bernackaitė, Dominyka Kievinaitė, and Jonas Šiaulys. "A Lundberg-type inequality for an inhomogeneous renewal risk model." Modern Stochastics: Theory and Applications 2, no. 2 (July 31, 2015): 173–84. http://dx.doi.org/10.15559/15-vmsta30.

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12

Delsing, Guusje, and Michel Mandjes. "A transient Cramér–Lundberg model with applications to credit risk." Journal of Applied Probability 58, no. 3 (September 2021): 721–45. http://dx.doi.org/10.1017/jpr.2020.114.

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AbstractThis paper considers a variant of the classical Cramér–Lundberg model that is particularly appropriate in the credit context, with the distinguishing feature that it corresponds to a finite number of obligors. The focus is on computing the ruin probability, i.e. the probability that the initial reserve, increased by the interest received from the obligors and decreased by the losses due to defaults, drops below zero. As well as an exact analysis (in terms of transforms) of this ruin probability, an asymptotic analysis is performed, including an efficient importance-sampling-based simulation approach.The base model is extended in multiple dimensions: (i) we consider a model in which there may, in addition, be losses that do not correspond to defaults, (ii) then we analyze a model in which the individual obligors are coupled via a regime switching mechanism, (iii) then we extend the model so that between the losses the reserve process behaves as a Brownian motion rather than a deterministic drift, and (iv) we finally consider a set-up with multiple groups of statistically identical obligors.

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13

Huang, Yujuan, and Wenguang Yu. "Studies on a Double Poisson-Geometric Insurance Risk Model with Interference." Discrete Dynamics in Nature and Society 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/128796.

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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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14

Jordanova, Pavlina K., and Milan Stehlík. "On multivariate modifications of Cramer–Lundberg risk model with constant intensities." Stochastic Analysis and Applications 36, no. 5 (May 31, 2018): 858–82. http://dx.doi.org/10.1080/07362994.2018.1471403.

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15

Schmidli, Hanspeter. "Estimation of the Lundberg coefficient for a Markov modulated risk model." Scandinavian Actuarial Journal 1997, no. 1 (January 1997): 48–57. http://dx.doi.org/10.1080/03461238.1997.10413977.

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16

Albrecher, Hansjörg, Jean-François Renaud, and Xiaowen Zhou. "A Lévy Insurance Risk Process with Tax." Journal of Applied Probability 45, no. 02 (June 2008): 363–75. http://dx.doi.org/10.1017/s0021900200004289.

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Using fluctuation theory, we solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Lévy risk process with tax payments of a loss-carry-forward type. We study arbitrary moments of the discounted total amount of tax payments and determine the surplus level to start taxation which maximises the expected discounted aggregate income for the tax authority in this model. The results considerably generalise those for the Cramér-Lundberg risk model with tax.

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17

Albrecher, Hansjörg, Jean-François Renaud, and Xiaowen Zhou. "A Lévy Insurance Risk Process with Tax." Journal of Applied Probability 45, no. 2 (June 2008): 363–75. http://dx.doi.org/10.1239/jap/1214950353.

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Using fluctuation theory, we solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Lévy risk process with tax payments of a loss-carry-forward type. We study arbitrary moments of the discounted total amount of tax payments and determine the surplus level to start taxation which maximises the expected discounted aggregate income for the tax authority in this model. The results considerably generalise those for the Cramér-Lundberg risk model with tax.

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18

Chan, Gary K. C., and Hailiang Yang. "UPPER BOUNDS FOR RUIN PROBABILITY UNDER TIME SERIES MODELS." Probability in the Engineering and Informational Sciences 20, no. 3 (June 1, 2006): 529–42. http://dx.doi.org/10.1017/s0269964806060323.

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In this article, we consider an insurance risk model where the claim and premium processes follow some time series models. We first consider the model proposed in Gerber [2,3]; then a model with dependent structure between premium and claim processes modeled by using Granger's causal model is considered. By using some martingale arguments, Lundberg-type upper bounds for the ruin probabilities under both models are obtained. Some special cases are discussed.

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19

Li, Yan, and Guoxin Liu. "Optimal Dividend and Capital Injection Strategies in the Cramér-Lundberg Risk Model." Mathematical Problems in Engineering 2015 (2015): 1–16. http://dx.doi.org/10.1155/2015/439537.

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We discuss the optimal dividend and capital injection strategies in the Cramér-Lundberg risk model. The value functionV(x)is defined by maximizing the discounted value of the dividend payment minus the penalized discounted capital injection until the time of ruin. It is shown thatV(x)can be characterized by the Hamilton-Jacobi-Bellman equation. We find the optimal dividend barrierb, the optimal upper capital injection barrier 0, and the optimal lower capital injection barrier-z*. In the case of exponential claim size especially, we give an explicit procedure to obtainb,-z*, and the value functionV(x).

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20

Ng, Andrew C. Y. "On the Upcrossing and Downcrossing Probabilities of a Dual Risk Model With Phase-Type Gains." ASTIN Bulletin 40, no. 1 (May 2010): 281–306. http://dx.doi.org/10.2143/ast.40.1.2049230.

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AbstractIn this paper, we consider the dual of the classical Cramér-Lundberg model when gains follow a phase-type distribution. By using the property of phase-type distribution, two pairs of upcrossing and downcrossing barrier probabilities are derived. Explicit formulas for the expected total discounted dividends until ruin and the Laplace transform of the time of ruin under a variety of dividend strategies can then be obtained without the use of Laplace transforms.

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21

Kuras, Tautvydas, Jonas Sprindys, and Jonas Šiaulys. "Martingale Approach to Derive Lundberg-Type Inequalities." Mathematics 8, no. 10 (October 11, 2020): 1742. http://dx.doi.org/10.3390/math8101742.

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In this paper, we find the upper bound for the tail probability Psupn⩾0∑I=1nξI>x with random summands ξ1,ξ2,… having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity ϱ1exp{−ϱ2x} with some positive constants ϱ1 and ϱ2. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.

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22

Liang, Xiaoqing, and Virginia R. Young. "Discounted probability of exponential parisian ruin: Diffusion approximation." Journal of Applied Probability 59, no. 1 (February 18, 2022): 17–37. http://dx.doi.org/10.1017/jpr.2021.36.

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AbstractWe analyze the discounted probability of exponential Parisian ruin for the so-called scaled classical Cramér–Lundberg risk model. As in Cohen and Young (2020), we use the comparison method from differential equations to prove that the discounted probability of exponential Parisian ruin for the scaled classical risk model converges to the corresponding discounted probability for its diffusion approximation, and we derive the rate of convergence.

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23

Kartashov, M. V., and V. V. Golomozyĭ. "Some inequalities for the risk function in the time and space nonhomogeneous Cramér–Lundberg risk model." Theory of Probability and Mathematical Statistics 98 (August 19, 2019): 243–54. http://dx.doi.org/10.1090/tpms/1074.

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24

Romera, R., and W. Runggaldier. "Ruin Probabilities in a Finite-Horizon Risk Model with Investment and Reinsurance." Journal of Applied Probability 49, no. 04 (December 2012): 954–66. http://dx.doi.org/10.1017/s0021900200012808.

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A finite-horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Our setting is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time semi-Markov process which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability.

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25

Romera, R., and W. Runggaldier. "Ruin Probabilities in a Finite-Horizon Risk Model with Investment and Reinsurance." Journal of Applied Probability 49, no. 4 (December 2012): 954–66. http://dx.doi.org/10.1239/jap/1354716650.

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A finite-horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Our setting is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time semi-Markov process which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability.

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26

Boxma, Onno J., Esther Frostig, and David Perry. "A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function." Journal of Applied Probability 54, no. 1 (March 2017): 267–85. http://dx.doi.org/10.1017/jpr.2016.99.

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AbstractWe consider a Cramér–Lundberg insurance risk process with the added feature of reinsurance. If an arriving claim finds the reserve below a certain threshold γ, or if it would bring the reserve below that level, then a reinsurer pays part of the claim. Using fluctuation theory and the theory of scale functions of spectrally negative Lévy processes, we derive expressions for the Laplace transform of the time to ruin and of the joint distribution of the deficit at ruin and the surplus before ruin. We specify these results in much more detail for the threshold set-up in the case of proportional reinsurance.

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27

Malinovskii, Vsevolod K. "Improved asymptotic upper bounds on the ruin capital in the Lundberg model of risk." Insurance: Mathematics and Economics 55 (March 2014): 301–9. http://dx.doi.org/10.1016/j.insmatheco.2013.12.004.

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28

Ng, Andrew C. Y., and Hailiang Yang. "Lundberg-type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin under a Markov-modulated Risk Model." ASTIN Bulletin 35, no. 02 (November 2005): 351–61. http://dx.doi.org/10.2143/ast.35.2.2003457.

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In this paper, we consider a Markov-modulated risk model (also called Markovian regime switching insurance risk model). Follow Asmussen (2000, 2003), by using the theory of Markov additive process, an exponential martingale is constructed and Lundberg-type upper bounds for the joint distribution of surplus immediately before and at ruin are obtained. As a natural corollary, bounds for the distribution of the deficit at ruin are obtained. We also present some numerical results to illustrate the tightness of the bound obtained in this paper.

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29

Ng, Andrew C. Y., and Hailiang Yang. "Lundberg-type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin under a Markov-modulated Risk Model." ASTIN Bulletin 35, no. 2 (November 2005): 351–61. http://dx.doi.org/10.1017/s0515036100014288.

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In this paper, we consider a Markov-modulated risk model (also called Markovian regime switching insurance risk model). Follow Asmussen (2000, 2003), by using the theory of Markov additive process, an exponential martingale is constructed and Lundberg-type upper bounds for the joint distribution of surplus immediately before and at ruin are obtained. As a natural corollary, bounds for the distribution of the deficit at ruin are obtained. We also present some numerical results to illustrate the tightness of the bound obtained in this paper.

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30

Avram, Florin, Andras Horváth, Serge Provost, and Ulyses Solon. "On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes." Risks 7, no. 4 (December 11, 2019): 121. http://dx.doi.org/10.3390/risks7040121.

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This paper considers the Brownian perturbed Cramér–Lundberg risk model with a dividends barrier. We study various types of Padé approximations and Laguerre expansions to compute or approximate the scale function that is necessary to optimize the dividends barrier. We experiment also with a heavy-tailed claim distribution for which we apply the so-called “shifted” Padé approximation.

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31

Fang, Ying, and Zhongfeng Qu. "Optimal Dividend and Capital Injection Strategies for a Risk Model under Force of Interest." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/750547.

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As a generalization of the classical Cramér-Lundberg risk model, we consider a risk model including a constant force of interest in the present paper. Most optimal dividend strategies which only consider the processes modeling the surplus of a risk business are absorbed at 0. However, in many cases, negative surplus does not necessarily mean that the business has to stop. Therefore, we assume that negative surplus is not allowed and the beneficiary of the dividends is required to inject capital into the insurance company to ensure that its risk process stays nonnegative. For this risk model, we show that the optimal dividend strategy which maximizes the discounted dividend payments minus the penalized discounted capital injections is a threshold strategy for the case of the dividend payout rate which is bounded by some positive constant and the optimal injection strategy is to inject capitals immediately to make the company's assets back to zero when the surplus of the company becomes negative.

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32

ABDERRAHIM, EL ATTAR, EL HACHLOUFI MOSTAFA, and GUENNOUN ZINE EL ABIDINE. "AN INCLUSIVE CRITERION FOR AN OPTIMAL CHOICE OF REINSURANCE." Annals of Financial Economics 12, no. 04 (December 2017): 1750018. http://dx.doi.org/10.1142/s201049521750018x.

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In this paper, we propose an inclusive model which allows to improve the results obtained in the literature with regard to the criteria set by the insurers such as, maximizing the expected technical benefit under the variance constraint (mean-variance), minimizing the probability of ruin and minimizing risk measures. In this model, we determine the optimal reinsurance treaty parameter that minimizes both the risk and the probability of ruin (by maximizing the Lundberg adjustment coefficient) under the constraint of the technical benefit which must also be maximal, based on the conditional tail variance (CTV) risk measure. Thus, we have developed an optimization procedure based on the augmented Lagrangian and genetic algorithms, in order to solve the optimization program of this model.

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33

Ji, Lanpeng, and Chunsheng Zhang. "Analysis of the multiple roots of the Lundberg fundamental equation in the PH (n) risk model." Applied Stochastic Models in Business and Industry 28, no. 1 (April 19, 2011): 73–90. http://dx.doi.org/10.1002/asmb.899.

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34

Kyprianou, Andreas E., and Curdin Ott. "Spectrally Negative Lévy Processes Perturbed by Functionals of their Running Supremum." Journal of Applied Probability 49, no. 4 (December 2012): 1005–14. http://dx.doi.org/10.1239/jap/1354716654.

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In the setting of the classical Cramér–Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if X = {Xt: t≥ 0} represents the Cramér–Lundberg process and, for all t≥ 0, St=sup_{s≤ t}Xs, then Albrecher and Hipp studied Xt - γ St,t≥ 0, where γ∈(0,1) is the rate at which tax is paid. This model has been generalised to the setting that X is a spectrally negative Lévy process by Albrecher, Renaud and Zhou (2008). Finally, Kyprianou and Zhou (2009) extended this model further by allowing the rate at which tax is paid with respect to the process S = {St: t≥ 0} to vary as a function of the current value of S. Specifically, they considered the so-called perturbed spectrally negative Lévy process, Ut:=Xt -∫(0,t]γ(S_u)dSu,t≥ 0, under the assumptions that γ:[0,∞)→ [0,1) and ∫0∞ (1-γ(s))d s =∞. In this article we show that a number of the identities in Kyprianou and Zhou (2009) are still valid for a much more general class of rate functions γ:[0,∞)→∝. Moreover, we show that, with appropriately chosen γ, the perturbed process can pass continuously (i.e. creep) into (-∞, 0) in two different ways.

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35

Osatakul, Dhiti, and Xueyuan Wu. "Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment." Risks 9, no. 1 (January 14, 2021): 26. http://dx.doi.org/10.3390/risks9010026.

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In this paper we consider a discrete-time risk model, which allows the premium to be adjusted according to claims experience. This model is inspired by the well-known bonus-malus system in the non-life insurance industry. Two strategies of adjusting periodic premiums are considered: aggregate claims or claim frequency. Recursive formulae are derived to compute the finite-time ruin probabilities, and Lundberg-type upper bounds are also derived to evaluate the ultimate-time ruin probabilities. In addition, we extend the risk model by considering an external Markovian environment in which the claims distributions are governed by an external Markov process so that the periodic premium adjustments vary when the external environment state changes. We then study the joint distribution of premium level and environment state at ruin given ruin occurs. Two numerical examples are provided at the end of this paper to illustrate the impact of the initial external environment state, the initial premium level and the initial surplus on the ruin probability.

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36

Kyprianou, Andreas E., and Curdin Ott. "Spectrally Negative Lévy Processes Perturbed by Functionals of their Running Supremum." Journal of Applied Probability 49, no. 04 (December 2012): 1005–14. http://dx.doi.org/10.1017/s0021900200012845.

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In the setting of the classical Cramér–Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if X = {X t : t≥ 0} represents the Cramér–Lundberg process and, for all t≥ 0, S t =sup_{s≤ t}X s , then Albrecher and Hipp studied X t - γ S t ,t≥ 0, where γ∈(0,1) is the rate at which tax is paid. This model has been generalised to the setting that X is a spectrally negative Lévy process by Albrecher, Renaud and Zhou (2008). Finally, Kyprianou and Zhou (2009) extended this model further by allowing the rate at which tax is paid with respect to the process S = {S t : t≥ 0} to vary as a function of the current value of S. Specifically, they considered the so-called perturbed spectrally negative Lévy process, U t :=X t -∫(0,t]γ(S_u)dS u ,t≥ 0, under the assumptions that γ:[0,∞)→ [0,1) and ∫0 ∞ (1-γ(s))d s =∞. In this article we show that a number of the identities in Kyprianou and Zhou (2009) are still valid for a much more general class of rate functions γ:[0,∞)→∝. Moreover, we show that, with appropriately chosen γ, the perturbed process can pass continuously (i.e. creep) into (-∞, 0) in two different ways.

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Kasumo, Christian, Juma Kasozi, and Dmitry Kuznetsov. "On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance." Journal of Applied Mathematics 2018 (2018): 1–11. http://dx.doi.org/10.1155/2018/9180780.

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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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Potocký, Rastislav, and Milan Stehlík. "Statistical analysis of mixtures underlying probability of ruin." Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis 57, no. 6 (2009): 209–14. http://dx.doi.org/10.11118/actaun200957060209.

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If the hypothesis on exponentially distributed claims in a risk (or surplus) model is untenable then, in many cases, the assumption that they are mixtures of two (or more) exponentials is a suitable substitute. In the first part of the paper tests of homogeneity for exponentially distributed claims are discussed and their properties are stated. The statistical properties of parameter estimations for such claims are also mentioned. In the second part the classical Cramer-Lundberg ruin model is discussed when claims are distributed as mixtures of exponentials. Our attention is focussed primarily on assesment of accuracy of approximations obtained. Then our results are compared to those already known.

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Asmussen, Soren, Florin Avram, and Miguel Usabel. "Erlangian Approximations for Finite-Horizon Ruin Probabilities." ASTIN Bulletin 32, no. 2 (November 2002): 267–81. http://dx.doi.org/10.2143/ast.32.2.1029.

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AbstractFor the Cramér-Lundberg risk model with phase-type claims, it is shown that the probability of ruin before an independent phase-type time H coincides with the ruin probability in a certain Markovian fluid model and therefore has an matrix-exponential form. When H is exponential, this yields in particular a probabilistic interpretation of a recent result of Avram & Usabel. When H is Erlang, the matrix algebra takes a simple recursive form, and fixing the mean of H at T and letting the number of stages go to infinity yields a quick approximation procedure for the probability of ruin before time T. Numerical examples are given, including a combination with extrapolation.

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Albrecher, Hansjörg, Jürgen Hartinger, and Stefan Thonhauser. "On Exact Solutions for Dividend Strategies of Threshold and Linear Barrier Type in a Sparre Andersen Model." ASTIN Bulletin 37, no. 02 (November 2007): 203–33. http://dx.doi.org/10.2143/ast.37.2.2024065.

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For the classical Cramér-Lundberg risk model, a dividend strategy of threshold type has recently been suggested in the literature. This strategy consists of paying out part of the premium income as dividends to shareholders whenever the free surplus is above a given threshold level. In contrast to the well-known horizontal barrier strategy, the threshold strategy can lead to a positive infinite-horizon survival probability, with reduced profit in terms of dividend payments. In this paper we extend several of these results to a Sparre Andersen model with generalized Erlang(n)-distributed interclaim times. Furthermore, we compare the performance of the threshold strategy to a linear dividend barrier model. In particular, (partial) integro-differential equations for the corresponding ruin probabilities and expected discounted dividend payments are provided for both models and explicitly solved for n = 2 and exponentially distributed claim amounts. Finally, the explicit solutions are used to identify parameter sets for which one strategy outperforms the other and vice versa.

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Albrecher, Hansjörg, Jürgen Hartinger, and Stefan Thonhauser. "On Exact Solutions for Dividend Strategies of Threshold and Linear Barrier Type in a Sparre Andersen Model." ASTIN Bulletin 37, no. 2 (November 2007): 203–33. http://dx.doi.org/10.1017/s0515036100014847.

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For the classical Cramér-Lundberg risk model, a dividend strategy of threshold type has recently been suggested in the literature. This strategy consists of paying out part of the premium income as dividends to shareholders whenever the free surplus is above a given threshold level. In contrast to the well-known horizontal barrier strategy, the threshold strategy can lead to a positive infinite-horizon survival probability, with reduced profit in terms of dividend payments. In this paper we extend several of these results to a Sparre Andersen model with generalized Erlang(n)-distributed interclaim times. Furthermore, we compare the performance of the threshold strategy to a linear dividend barrier model. In particular, (partial) integro-differential equations for the corresponding ruin probabilities and expected discounted dividend payments are provided for both models and explicitly solved for n = 2 and exponentially distributed claim amounts. Finally, the explicit solutions are used to identify parameter sets for which one strategy outperforms the other and vice versa.

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Choi, Michael C. H., and Eric C. K. Cheung. "On the expected discounted dividends in the Cramér–Lundberg risk model with more frequent ruin monitoring than dividend decisions." Insurance: Mathematics and Economics 59 (November 2014): 121–32. http://dx.doi.org/10.1016/j.insmatheco.2014.08.009.

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Yang, Peng. "Optimal Reinsurance-Investment Problem under Mean-Variance Criterion with n Risky Assets." Discrete Dynamics in Nature and Society 2020 (June 1, 2020): 1–16. http://dx.doi.org/10.1155/2020/6489532.

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Based on the mean-variance criterion, this paper investigates the continuous-time reinsurance and investment problem. The insurer’s surplus process is assumed to follow Cramér–Lundberg model. The insurer is allowed to purchase reinsurance for reducing claim risk. The reinsurance pattern that the insurer adopts is combining proportional and excess of loss reinsurance. In addition, the insurer can invest in financial market to increase his wealth. The financial market consists of one risk-free asset and n correlated risky assets. The objective is to minimize the variance of the terminal wealth under the given expected value of the terminal wealth. By applying the principle of dynamic programming, we establish a Hamilton–Jacobi–Bellman (HJB) equation. Furthermore, we derive the explicit solutions for the optimal reinsurance-investment strategy and the corresponding efficient frontier by solving the HJB equation. Finally, numerical examples are provided to illustrate how the optimal reinsurance-investment strategy changes with model parameters.

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Brachetta, Matteo, and Claudia Ceci. "Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences." Mathematics 9, no. 4 (February 3, 2021): 295. http://dx.doi.org/10.3390/math9040295.

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We investigate an optimal reinsurance problem for an insurance company taking into account subscription costs: that is, a constant fixed cost is paid when the reinsurance contract is signed. Differently from the classical reinsurance problem, where the insurer has to choose an optimal retention level according to some given criterion, in this paper, the insurer needs to optimally choose both the starting time of the reinsurance contract and the retention level to apply. The criterion is the maximization of the insurer’s expected utility of terminal wealth. This leads to a mixed optimal control/optimal stopping time problem, which is solved by a two-step procedure: first considering the pure-reinsurance stochastic control problem and next discussing a time-inhomogeneous optimal stopping problem with discontinuous reward. Using the classical Cramér–Lundberg approximation risk model, we prove that the optimal strategy is deterministic and depends on the model parameters. In particular, we show that there exists a maximum fixed cost that the insurer is willing to pay for the contract activation. Finally, we provide some economical interpretations and numerical simulations.

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Yiou, Pascal, and Nicolas Viovy. "Modelling forest ruin due to climate hazards." Earth System Dynamics 12, no. 3 (September 21, 2021): 997–1013. http://dx.doi.org/10.5194/esd-12-997-2021.

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Abstract. Estimating the risk of forest collapse due to extreme climate events is one of the challenges of adapting to climate change. We adapt a concept from ruin theory, which is widely used in econometrics and the insurance industry, to design a growth–ruin model for trees which accounts for climate hazards that can jeopardize tree growth. This model is an elaboration of a classical Cramer–Lundberg ruin model that is used in the insurance industry. The model accounts for the interactions between physiological parameters of trees and the occurrence of climate hazards. The physiological parameters describe interannual growth rates and how trees react to hazards. The hazard parameters describe the probability distributions of the occurrence and intensity of climate events. We focus on a drought–heatwave hazard. The goal of the paper is to determine the dependence of the forest ruin and average growth probability distributions on physiological and hazard parameters. Using extensive Monte Carlo experiments, we show the existence of a threshold in the frequency of hazards beyond which forest ruin becomes certain to occur within a centennial horizon. We also detect a small effect of the strategies used to cope with hazards. This paper is a proof of concept for the quantification of forest collapse under climate change.

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Afonso, Lourdes B., Rui M. R. Cardoso, Alfredo D. Egídio dos Reis, and Gracinda Rita Guerreiro. "MEASURING THE IMPACT OF A BONUS-MALUS SYSTEM IN FINITE AND CONTINUOUS TIME RUIN PROBABILITIES FOR LARGE PORTFOLIOS IN MOTOR INSURANCE." ASTIN Bulletin 47, no. 2 (March 21, 2017): 417–35. http://dx.doi.org/10.1017/asb.2017.3.

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AbstractMotor insurance is a very competitive business where insurers operate with quite large portfolios, often decisions must be taken under short horizons and therefore ruin probabilities should be calculated in finite time. The probability of ruin, in continuous and finite time, is numerically evaluated under the classical Cramér–Lundberg risk process framework for a large motor insurance portfolio, where we allow for a posteriori premium adjustments, according to the claim record of each individual policyholder. Focusing on the classical model for bonus-malus systems, we propose that the probability of ruin can be interpreted as a measure to decide between different bonus-malus scales or even between different bonus-malus rules. In our work, the required initial surplus can also be evaluated. We consider an application of a bonus-malus system for motor insurance to study the impact of experience rating in ruin probabilities. For that, we used a real commercial scale of an insurer operating in the Portuguese market, and we also work on various well-known optimal bonus-malus scales estimated with real data from that insurer. Results involving these scales are discussed.

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Zhou, Qianqian, Alexander Sakhanenko, and Junyi Guo. "Lundberg-type inequalities for non-homogeneous risk models." Stochastic Models 36, no. 4 (October 1, 2020): 661–80. http://dx.doi.org/10.1080/15326349.2020.1835490.

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Xiong, Sheng, and Wei-Shih Yang. "Ruin probability in the Cramér–Lundberg model with risky investments." Stochastic Processes and their Applications 121, no. 5 (May 2011): 1125–37. http://dx.doi.org/10.1016/j.spa.2011.01.008.

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Cai, Jun. "Ruin probabilities with dependent rates of interest." Journal of Applied Probability 39, no. 02 (June 2002): 312–23. http://dx.doi.org/10.1017/s0021900200022531.

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In this paper, we study ruin probabilities in two generalized risk models. The effects of timing of payments and interest on the ruin probabilities in the models are considered. The rates of interest are assumed to have a dependent autoregressive structure. Generalized Lundberg inequalities for the ruin probabilities are derived by a renewal recursive technique. An illustrative application is given to the compound binomial risk process.

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Cai, Jun. "Ruin probabilities with dependent rates of interest." Journal of Applied Probability 39, no. 2 (June 2002): 312–23. http://dx.doi.org/10.1239/jap/1025131428.

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In this paper, we study ruin probabilities in two generalized risk models. The effects of timing of payments and interest on the ruin probabilities in the models are considered. The rates of interest are assumed to have a dependent autoregressive structure. Generalized Lundberg inequalities for the ruin probabilities are derived by a renewal recursive technique. An illustrative application is given to the compound binomial risk process.

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