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Olena Ragulina - One of the best experts on this subject based on the ideXlab platform.
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Classical Results on the Ruin Probabilities
Ruin Probabilities, 2020Co-Authors: Yuliya Mishura, Olena RagulinaAbstract:In this chapter, we formulate some basic results concerning ruin probabilities in the Classical Risk Model and the Risk Model with stochastic premiums.
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Risk Models with Investments in Risk-Free and Risky Assets
Ruin Probabilities, 2020Co-Authors: Yuliya Mishura, Olena RagulinaAbstract:In this chapter, we deal with generalizations of the Classical Risk Model and the Risk Model with stochastic premiums where an insurance company invests all surplus in Risk-free and Risky assets proportionally. The price of the Risky asset follows a jump process. We get upper and lower bounds for the infinite-horizon survival probability and investigate the continuity and differentiability of the infinite- and finite-horizon survival probabilities in the generalization of the Classical Risk Model. Moreover, we extend these results to the generalization of the Risk Model with stochastic premiums. Finally, we obtain relations connecting accuracy and reliability of uniform approximations of the survival probabilities by their statistical estimates.
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Classical Risk Model with investments in a Risk free asset
Ruin Probabilities#R##N#Smoothness Bounds Supermartingale Approach, 2017Co-Authors: Yuliya Mishura, Olena RagulinaAbstract:In this chapter, we deal with a generalization of the Classical Risk Model where an insurance company invests all surplus in a Risk-free asset. We investigate the continuity and differentiability of the infinite- and finitehorizon survival probabilities in detail. Moreover, we derive integrodifferential equations for these functions and get bounds for their derivatives w.r.t. the initial surplus.
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Classical Risk Model with a franchise and a liability limit
Ruin Probabilities#R##N#Smoothness Bounds Supermartingale Approach, 2017Co-Authors: Yuliya Mishura, Olena RagulinaAbstract:In this chapter, we deal with the Classical Risk Model under the additional assumption that an insurance company applies a franchise and a liability limit. To be more precise, we consider three cases: the insurance company establishes a franchise only, a liability limit only and both a franchise and a liability limit. Assuming that claim sizes are exponentially distributed we find analytic expressions for the infinite-horizon survival probabilities in these cases. The expressions turn out different on certain intervals. Moreover, we investigate how a franchise and a liability limit change the survival probability for small and large enough initial surpluses.
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optimal control by the franchise and deductible amounts in the Classical Risk Model
Ruin Probabilities#R##N#Smoothness Bounds Supermartingale Approach, 2017Co-Authors: Yuliya Mishura, Olena RagulinaAbstract:In this chapter, we consider the Classical Risk Model where an insurance company is able to adjust a franchise amount continuously. The problem of optimal control by the franchise amount is solved from the viewpoint of survival probability maximization. We derive the Hamilton–Jacobi–Bellman equation for the optimal survival probability and prove the existence of the solution to this equation with certain properties. The verification theorem gives the connection between this solution and the optimal survival probability, which differ in a constant multiplier. Then, we concentrate on the case of exponentially distributed claim sizes. Finally, we extend these results to the problem of optimal control by a deductible amount.
David C.m. Dickson - One of the best experts on this subject based on the ideXlab platform.
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Some Finite Time Ruin Problems
Annals of Actuarial Science, 2020Co-Authors: David C.m. DicksonAbstract:ABSTRACTIn the Classical Risk Model, we use probabilistic arguments to write down expressions in terms of the density function of aggregate claims for joint density functions involving the time to ruin, the deficit at ruin and the surplus prior to ruin. We give some applications of these formulae in the cases when the individual claim amount distribution is exponential and Erlang(2).
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Some Explicit Solutions for the Joint Density of the Time of Ruin and the Deficit at Ruin
ASTIN Bulletin, 2020Co-Authors: David C.m. DicksonAbstract:Using probabilistic arguments we obtain an integral expression for the joint density of the time of ruin and the deficit at ruin. For the Classical Risk Model, we obtain the bivariate Laplace transform of this joint density and invert it in the cases of individual claims distributed as Erlang(2) and as a mixture of two exponential distributions. As a consequence, we obtain explicit solutions for the density of the time of ruin.
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Gerber–Shiu analysis of a Risk Model with capital injections
European Actuarial Journal, 2016Co-Authors: David C.m. Dickson, Marjan QazviniAbstract:We consider the Risk Model with capital injections studied by Nie et al. (Ann Actuar Sci 5:195–209, 2011 ; Scand Actuar J 2015:301–318, 2015 ). We construct a Gerber–Shiu function and show that whilst this tool is not efficient for finding the ultimate ruin probability, it provides an effective way of studying ruin related quantities in finite time. In particular, we find a general expression for the joint distribution of the time of ruin and the number of claims until ruin, and find an extension of Prabhu’s (Ann Math Stat 32:757–764, 1961 ) formula for the finite time survival probability in the Classical Risk Model. We illustrate our results in the case of exponentially distributed claims and obtain some interesting identities. In particular, we generalise results from the Classical Risk Model and prove the identity of two known formulae for that Model.
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the joint distribution of the time to ruin and the number of claims until ruin in the Classical Risk Model
Insurance Mathematics & Economics, 2012Co-Authors: David C.m. DicksonAbstract:We use probabilistic arguments to derive an expression for the joint density of the time to ruin and the number of claims until ruin in the Classical Risk Model. From this we obtain a general expression for the probability function of the number of claims until ruin. We also consider the moments of the number of claims until ruin and illustrate our results in the case of exponentially distributed individual claims. Finally, we briefly discuss joint distributions involving the surplus prior to ruin and deficit at ruin.
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the density of the time to ruin in the Classical poisson Risk Model
Astin Bulletin, 2005Co-Authors: David C.m. Dickson, Gordon E WillmotAbstract:We derive an expression for the density of the time to ruin in the Classical Risk Model by inverting its Laplace transform. We then apply the result when the individual claim amount distribution is a mixed Erlang distribution, and show how finite time ruin probabilities can be calculated in this case.
Huayue Zhang - One of the best experts on this subject based on the ideXlab platform.
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insurance control for Classical Risk Model with fractional brownian motion perturbation
Statistics & Probability Letters, 2009Co-Authors: Huayue Zhang, A M ZhouAbstract:In the paper, we consider a Classical Risk Model that is perturbed by a standard fractional Brownian motion with Hurst parameter . The customers' input may be considered as a control parameter which allows the firm to reach a desired target at a specified time. By using the completion of squares method, we obtain an expression of the optimal value function and the corresponding optimal control policy.
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dynamic mean variance optimization under Classical Risk Model with fractional brownian motion perturbation
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2008Co-Authors: Huayue ZhangAbstract:In this paper, we apply the completion of squares method to study the optimal investment problem under mean-variance criteria for an insurer. The insurer's Risk process is Modelled by a Classical Risk process that is perturbed by a standard fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). By virtue of an auxiliary process, the efficient strategy and efficient frontier are obtained. Moreover, when H → 1/2+ the results converge to the corresponding (known) results for standard Brownian motion.
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the gerber shiu discounted penalty function for Classical Risk Model with a two step premium rate
Statistics & Probability Letters, 2006Co-Authors: Huayue Zhang, Ming ZhouAbstract:The paper studies the expected value of a discounted penalty function for a Classical Risk Model with a two-step premium rate. In this Model, we firstly derive and solve an integro-differential equation for the Gerber-Shiu discounted penalty function, then use this result to obtain the expressions of ruin probability and the joint distribution of the surplus immediately before ruin and the deficit at ruin.
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The Gerber–Shiu discounted penalty function for Classical Risk Model with a two-step premium rate
Statistics & Probability Letters, 2006Co-Authors: Huayue Zhang, Ming ZhouAbstract:The paper studies the expected value of a discounted penalty function for a Classical Risk Model with a two-step premium rate. In this Model, we firstly derive and solve an integro-differential equation for the Gerber-Shiu discounted penalty function, then use this result to obtain the expressions of ruin probability and the joint distribution of the surplus immediately before ruin and the deficit at ruin.
Zhimin Zhang - One of the best experts on this subject based on the ideXlab platform.
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estimating the discounted density of the deficit at ruin by fourier cosine series expansion
Statistics & Probability Letters, 2019Co-Authors: Yang Yang, Wen Su, Zhimin ZhangAbstract:Abstract In this paper, we study the statistical estimation of the discounted density of the deficit at ruin in the Classical Risk Model. The estimator is constructed by the two-dimensional Fourier cosine series expansion. It is shown that the estimator is easily computed and has fast convergence rate. Some simulation results are presented to show that the estimator performs very well when the sample size is finite.
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a new efficient method for estimating the gerber shiu function in the Classical Risk Model
Scandinavian Actuarial Journal, 2018Co-Authors: Zhimin Zhang, Wen SuAbstract:In this paper, we propose a new efficient method for estimating the Gerber–Shiu discounted penalty function in the Classical Risk Model. We develop the Gerber–Shiu function on the Laguerre basis, a...
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A new efficient method for estimating the Gerber–Shiu function in the Classical Risk Model
Scandinavian Actuarial Journal, 2017Co-Authors: Zhimin Zhang, Wen SuAbstract:In this paper, we propose a new efficient method for estimating the Gerber–Shiu discounted penalty function in the Classical Risk Model. We develop the Gerber–Shiu function on the Laguerre basis, a...
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nonparametric estimation of the finite time ruin probability in the Classical Risk Model
Scandinavian Actuarial Journal, 2017Co-Authors: Zhimin ZhangAbstract:In this paper, we consider the nonparametric estimation of the finite time ruin probability in the Classical Risk Model. Suppose that the individual claim size distribution is unknown, but a random sample of claims is available. We construct the estimator by double Fourier transform. The asymptotic properties of the estimator are studied under large sample setting, and show that an almost convergence rate can be obtained. Some simulation examples are also provided to illustrate the performance of the estimator under finite sample size setting.
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on a nonparametric estimator for ruin probability in the Classical Risk Model
Scandinavian Actuarial Journal, 2014Co-Authors: Zhimin Zhang, Hailiang Yang, Hu YangAbstract:In this paper, we present a nonparametric estimator for ruin probability in the Classical Risk Model with unknown claim size distribution. We construct the estimator by Fourier inversion and kernel density estimation method. Under some conditions imposed on the kernel, bandwidth and claim size density, we present some large sample properties of the estimator. Some simulation studies are also given to show the finite sample performance of the estimator.
Wen Guang Yu - One of the best experts on this subject based on the ideXlab platform.
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improvement to the expected discounted penalty function for a Classical Risk Model with a threshold dividend strategy
Applied Mechanics and Materials, 2010Co-Authors: Wen Guang YuAbstract:In this paper, we study the expected discounted penalty function for a Classical Risk Model in which a threshold dividend strategy is used for a Classical Risk Model and the discount interest force process is not a constant, but a stochastic process driven by Poisson process and Wiener process. In this Model, we derive and solve an integro-differential equation for the expected discounted penalty function.
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The Gerber-Shiu Discounted Penalty Function with Stochastic Interest Force under a Two-Step Premium Rate Risk Model
2009 International Conference on Information Management Innovation Management and Industrial Engineering, 2009Co-Authors: Yujuan Huang, Wen Guang YuAbstract:In this paper, we consider the Gerber-Shiu discounted penalty function for a Classical Risk Model with a two-step premium rate and a linear dividend barrier. An integro-differential equation for the Gerber-Shiu discounted penalty function under stochastic interest force is derived and solved, then the Lundberg fundamental equation is given also.