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Hu Yang - One of the best experts on this subject based on the ideXlab platform.

  • on a nonparametric estimator for the finite time survival probability with zero Initial Surplus
    Acta Mathematicae Applicatae Sinica, 2016
    Co-Authors: Zhimin Zhang, Hailiang Yang, Hu Yang
    Abstract:

    In this paper, we consider the estimation of the finite time survival probability in the classical risk model when the Initial Surplus is zero. We construct a nonparametric estimator by Fourier inversion and kernel density estimation method. Under some mild assumptions imposed on the kernel, bandwidth and claim size density, we derive the order of the bias and variance, and show that the estimator has asymptotic normality property. Some simulation studies show that the estimator performs quite well in the finite sample setting.

  • On the absolute ruin in a map risk model with debit interest
    Advances in Applied Probability, 2011
    Co-Authors: Zhimin Zhang, Hailiang Yang, Hu Yang
    Abstract:

    In this paper we consider a risk model where claims arrive according to a Markovian arrival process (MAP). When the Surplus becomes negative or the insurer is in deficit, the insurer could borrow money at a constant debit interest rate to repay the claims. We derive the integro-differential equations satisfied by the discounted penalty functions and discuss the solutions. A matrix renewal equation is obtained for the discounted penalty function provided that the Initial Surplus is nonnegative. Based on this matrix renewal equation, we present some asymptotic formulae for the discounted penalty functions when the claim size distributions are heavy tailed.

  • Ruin problems in a discrete Markov risk model
    Statistics & Probability Letters, 2009
    Co-Authors: Hu Yang, Zhimin Zhang, Chunmei Lan
    Abstract:

    Abstract In this paper, we extend the compound binomial risk model to a Markov dependent model in which the claim occurrence and the claim amount are both regulated by a discrete time Markov process. The explicit expression for the “discounted” joint probability function of the Surplus before ruin and the deficit at ruin is derived when the Initial Surplus u = 0 , and a recursive formula to calculate such “discounted” joint probability function when the Initial Surplus u > 0 is also obtained.

  • The Gerber-Shiu discounted penalty functions for a risk model with two classes of claims
    Journal of Computational and Applied Mathematics, 2009
    Co-Authors: Zhimin Zhang, Hu Yang
    Abstract:

    In this paper, we consider the ruin problems for a risk model involving two independent classes of insurance risks. We assume that the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. When the generalized Lundberg equation has distinct roots with positive real parts, both of the Gerber-Shiu discounted penalty functions with zero Initial Surplus and the Laplace transforms of the Gerber-Shiu discounted penalty functions are obtained. Finally, some explicit expressions for the Gerber-Shiu discounted penalty functions with positive Initial Surplus are given when the claim size distributions belong to the rational family.

Hailiang Yang - One of the best experts on this subject based on the ideXlab platform.

  • on a nonparametric estimator for the finite time survival probability with zero Initial Surplus
    Acta Mathematicae Applicatae Sinica, 2016
    Co-Authors: Zhimin Zhang, Hailiang Yang, Hu Yang
    Abstract:

    In this paper, we consider the estimation of the finite time survival probability in the classical risk model when the Initial Surplus is zero. We construct a nonparametric estimator by Fourier inversion and kernel density estimation method. Under some mild assumptions imposed on the kernel, bandwidth and claim size density, we derive the order of the bias and variance, and show that the estimator has asymptotic normality property. Some simulation studies show that the estimator performs quite well in the finite sample setting.

  • On the absolute ruin in a map risk model with debit interest
    Advances in Applied Probability, 2011
    Co-Authors: Zhimin Zhang, Hailiang Yang, Hu Yang
    Abstract:

    In this paper we consider a risk model where claims arrive according to a Markovian arrival process (MAP). When the Surplus becomes negative or the insurer is in deficit, the insurer could borrow money at a constant debit interest rate to repay the claims. We derive the integro-differential equations satisfied by the discounted penalty functions and discuss the solutions. A matrix renewal equation is obtained for the discounted penalty function provided that the Initial Surplus is nonnegative. Based on this matrix renewal equation, we present some asymptotic formulae for the discounted penalty functions when the claim size distributions are heavy tailed.

  • An elementary approach to discrete models of dividend strategies
    Insurance: Mathematics and Economics, 2010
    Co-Authors: Hans U Gerber, Elias S W Shiu, Hailiang Yang
    Abstract:

    Abstract The paper studies a discrete counterpart of Gerber et al. (2006) . The Surplus of an insurance company (before dividends) is modeled as a time-homogeneous Markov chain with possible changes of size + 1 , 0 , − 1 , − 2 , − 3 , … . If a barrier strategy is applied for paying dividends, it is shown that the dividends-penalty identity holds. The identity expresses the expected present value of a penalty at ruin in terms of the expected discounted dividends until ruin and the expected present value of the penalty at ruin if no dividends are paid. For the problem of maximizing the difference between the expected discounted dividends until ruin and the expected present value of the penalty at ruin, barrier strategies play a prominent role. In some cases an optimal dividend barrier exists. The paper discusses in detail the special case where the distribution of the change in Surplus does not depend on the current Surplus (so that in the absence of dividends the Surplus process has independent increments). A closed-form result for zero Initial Surplus is given, and it is shown how the relevant quantities can be calculated recursively. Finally, it is shown how optimal dividend strategies can be determined; typically, they are band strategies.

  • On the joint distribution of Surplus before and after ruin under a Markovian regime switching model
    Stochastic Processes and their Applications, 2006
    Co-Authors: Hailiang Yang
    Abstract:

    AbstractWe consider a Markovian regime switching insurance risk model (also called Markov-modulated risk model). The closed form solutions for the joint distribution of Surplus before and after ruin when the Initial Surplus is zero or when the claim size distributions are phase-type distributed are obtained

  • Approximations for moments of deficit at ruin with exponential and subexponential claims
    Statistics & Probability Letters, 2002
    Co-Authors: Yebin Cheng, Qihe Tang, Hailiang Yang
    Abstract:

    Consider a renewal insurance risk model with Initial Surplus u?0 and let Au denote the de"cit at the time ofruin. This paper investigates the asymptotic behavior ofthe moments of Au as u tends to in"nity. Under the assumption that the claim size is exponentially or subexponentially distributed, we obtain some

Zhimin Zhang - One of the best experts on this subject based on the ideXlab platform.

  • on a nonparametric estimator for the finite time survival probability with zero Initial Surplus
    Acta Mathematicae Applicatae Sinica, 2016
    Co-Authors: Zhimin Zhang, Hailiang Yang, Hu Yang
    Abstract:

    In this paper, we consider the estimation of the finite time survival probability in the classical risk model when the Initial Surplus is zero. We construct a nonparametric estimator by Fourier inversion and kernel density estimation method. Under some mild assumptions imposed on the kernel, bandwidth and claim size density, we derive the order of the bias and variance, and show that the estimator has asymptotic normality property. Some simulation studies show that the estimator performs quite well in the finite sample setting.

  • On the absolute ruin in a map risk model with debit interest
    Advances in Applied Probability, 2011
    Co-Authors: Zhimin Zhang, Hailiang Yang, Hu Yang
    Abstract:

    In this paper we consider a risk model where claims arrive according to a Markovian arrival process (MAP). When the Surplus becomes negative or the insurer is in deficit, the insurer could borrow money at a constant debit interest rate to repay the claims. We derive the integro-differential equations satisfied by the discounted penalty functions and discuss the solutions. A matrix renewal equation is obtained for the discounted penalty function provided that the Initial Surplus is nonnegative. Based on this matrix renewal equation, we present some asymptotic formulae for the discounted penalty functions when the claim size distributions are heavy tailed.

  • Ruin problems in a discrete Markov risk model
    Statistics & Probability Letters, 2009
    Co-Authors: Hu Yang, Zhimin Zhang, Chunmei Lan
    Abstract:

    Abstract In this paper, we extend the compound binomial risk model to a Markov dependent model in which the claim occurrence and the claim amount are both regulated by a discrete time Markov process. The explicit expression for the “discounted” joint probability function of the Surplus before ruin and the deficit at ruin is derived when the Initial Surplus u = 0 , and a recursive formula to calculate such “discounted” joint probability function when the Initial Surplus u > 0 is also obtained.

  • The Gerber-Shiu discounted penalty functions for a risk model with two classes of claims
    Journal of Computational and Applied Mathematics, 2009
    Co-Authors: Zhimin Zhang, Hu Yang
    Abstract:

    In this paper, we consider the ruin problems for a risk model involving two independent classes of insurance risks. We assume that the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. When the generalized Lundberg equation has distinct roots with positive real parts, both of the Gerber-Shiu discounted penalty functions with zero Initial Surplus and the Laplace transforms of the Gerber-Shiu discounted penalty functions are obtained. Finally, some explicit expressions for the Gerber-Shiu discounted penalty functions with positive Initial Surplus are given when the claim size distributions belong to the rational family.

Qihe Tang - One of the best experts on this subject based on the ideXlab platform.

Jun Cai - One of the best experts on this subject based on the ideXlab platform.

  • The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber–Shiu Discounted Penalty Function
    Methodology and Computing in Applied Probability, 2009
    Co-Authors: Jun Cai, Runhuan Feng, Gordon E. Willmot
    Abstract:

    We modify the compound Poisson Surplus model for an insurer by including liquid reserves and interest on the Surplus. When the Surplus of an insurer is below a fixed level, the Surplus is kept as liquid reserves, which do not earn interest. When the Surplus attains the level, the excess of the Surplus over the level will receive interest at a constant rate. If the level goes to infinity, the modified model is reduced to the classical compound Poisson risk model. If the level is set to zero, the modified model becomes the compound Poisson risk model with interest. We study ruin probability and other quantities related to ruin in the modified compound Poisson Surplus model by the Gerber–Shiu function and discuss the impact of interest and liquid reserves on the ruin probability, the deficit at ruin, and other ruin quantities. First, we derive a system of integro-differential equations for the Gerber–Shiu function. By solving the system of equations, we obtain the general solution for the Gerber–Shiu function. Then, we give the exact solutions for the Gerber–Shiu function when the Initial Surplus is equal to the liquid reserve level or equal to zero. These solutions are the key to the exact solution for the Gerber–Shiu function in general cases. As applications, we derive the exact solution for the zero discounted Gerber–Shiu function when claim sizes are exponentially distributed and the exact solution for the ruin probability when claim sizes have Erlang(2) distributions. Finally, we use numerical examples to illustrate the impact of interest and liquid reserves on the ruin probability.

  • The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber-Shiu Discounted Penalty Function
    Methodology and Computing in Applied Probability, 2007
    Co-Authors: Jun Cai, Runhuan Feng, Gordon E. Willmot
    Abstract:

    We modify the compound Poisson Surplus model for an insurer by including liquid reserves and interest on the Surplus. When the Surplus of an insurer is below a fixed level, the Surplus is kept as liquid reserves, which do not earn interest. When the Surplus attains the level, the excess of the Surplus over the level will receive interest at a constant rate. If the level goes to infinity, the modified model is reduced to the classical compound Poisson risk model. If the level is set to zero, the modified model becomes the compound Poisson risk model with interest. We study ruin probability and other quantities related to ruin in the modified compound Poisson Surplus model by the Gerber–Shiu function and discuss the impact of interest and liquid reserves on the ruin probability, the deficit at ruin, and other ruin quantities. First, we derive a system of integro-differential equations for the Gerber–Shiu function. By solving the system of equations, we obtain the general solution for the Gerber–Shiu function. Then, we give the exact solutions for the Gerber–Shiu function when the Initial Surplus is equal to the liquid reserve level or equal to zero. These solutions are the key to the exact solution for the Gerber–Shiu function in general cases. As applications, we derive the exact solution for the zero discounted Gerber–Shiu function when claim sizes are exponentially distributed and the exact solution for the ruin probability when claim sizes have Erlang(2) distributions. Finally, we use numerical examples to illustrate the impact of interest and liquid reserves on the ruin probability.

  • On the time value of absolute ruin with debit interest
    Advances in Applied Probability, 2007
    Co-Authors: Jun Cai
    Abstract:

    Assume that the Surplus of an insurer follows a compound Poisson Surplus process. When the Surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay the debts from her premium income. The negative Surplus may return to a positive level. However, when the negative Surplus is below a certain critical level, the Surplus is no longer able to be positive. Absolute ruin occurs at this moment. In this paper, we study absolute ruin questions by defining an expected discounted penalty function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the Surplus just before absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integro-differential equations satisfied by the function and obtain a defective renewal equation that links the integro-differential equations in the system. Second, we show that when the Initial Surplus goes to infinity, the absolute ruin probability and the classical ruin probability are asymptotically equal for heavy-tailed claims while the ratio of the absolute ruin probability to the classical ruin probability goes to a positive constant that is less than one for light-tailed claims. Finally, we give explicit expressions for the function for exponential claims.