This article deals with the problem of minimizing ruin probability under optimal control for the continuous-time compound binomial model with investment. The jump mechanism in our article is different from that of Liu...This article deals with the problem of minimizing ruin probability under optimal control for the continuous-time compound binomial model with investment. The jump mechanism in our article is different from that of Liu et al [4]. Comparing with [4], the introduction of the investment, and hence, the additional Brownian motion term, makes the problem technically challenging. To overcome this technical difficulty, the theory of change of measure is used and an exponential martingale is obtained by virtue of the extended generator. The ruin probability is minimized through maximizing adjustment coefficient in the sense of Lundberg bounds. At the same time, the optimal investment strategy is obtained.展开更多
We consider a risk model with a premium rate which varies with the level of free reserves. In this model, the occurrence of claims is described by a Cox process with Markov intensity process, and the influence of stoc...We consider a risk model with a premium rate which varies with the level of free reserves. In this model, the occurrence of claims is described by a Cox process with Markov intensity process, and the influence of stochastic factors is considered by adding a diffusion process. The integro-differential equation for the ruin probability is derived by a infinitesimal method. Key words ruin probability - variable premium rate - diffusion process - Markov intensity CLC number O 211.9 Foundation item: Supported by the National Natural Science Foundation of China (10071058, 70273029)展开更多
In this paper, a new risk model is studied in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is p-thinning ...In this paper, a new risk model is studied in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is p-thinning process. The integral representations of the survival probability are gotten. The explicit formula of the survival probability on the infinite interval is obtained in the special casc cxponential distribution.The Lundberg inequality and the common formula of the ruin probability are gotten in terms of some techniques from martingale theory.展开更多
This paper studies a Sparre Andersen negative risk sums model in which the distribution of "interclaim" time is that of a sum of n independent exponential random variables. Thus, the Erlang(n) model is a special c...This paper studies a Sparre Andersen negative risk sums model in which the distribution of "interclaim" time is that of a sum of n independent exponential random variables. Thus, the Erlang(n) model is a special case. On this basis the correlated negative risk sums process with the common Erlang process is considered. Integro-differential equations with boundary conditions for ψ(u) are given. For some special cases a closed-form expression for ψ(u) is derived.展开更多
Let R(t)=u+ct-∑ I=1^N(t) Xi,t≥0 be the renewal risk model, with Fx(x)being the distribution function of the claim amount X. Let ψ(u) be the ruin probability with initial surplus u. Under the condition of F...Let R(t)=u+ct-∑ I=1^N(t) Xi,t≥0 be the renewal risk model, with Fx(x)being the distribution function of the claim amount X. Let ψ(u) be the ruin probability with initial surplus u. Under the condition of Fx(x) ∈ S^*(γ),y ≥ 0, by the geometric sum method, we derive the local asymptotic behavior for ψ(u,u + z] for every 0 ( z ( oo, On one hand, the asymptotic behavior of ψ(u) can be derived from the result obtained. On the other hand, the result of this paper can be applied to the insurance risk management of an insurance company.展开更多
Consider a continuous-time renewal risk model, in which every main claim induces a delayed by-claim. Assume that the main claim sizes and the inter-arrival times form a sequence of identically distributed random pairs...Consider a continuous-time renewal risk model, in which every main claim induces a delayed by-claim. Assume that the main claim sizes and the inter-arrival times form a sequence of identically distributed random pairs, with each pair obeying a dependence structure, and so do the by-claim sizes and the delay times. Supposing that the main claim sizes with by-claim sizes form a sequence of dependent random variables with dominatedly varying tails, asymptotic estimates for the ruin probability of the surplus process are investigated, by establishing a weakly asymptotic formula, as the initial surplus tends to infinity.展开更多
Consider a discrete-time insurance risk model. Within period i, i≥ 1, Xi and Yi denote the net insurance loss and the stochastic discount factor of an insurer, respectively. Assume that {(Xi, Yi), i≥1) form a seq...Consider a discrete-time insurance risk model. Within period i, i≥ 1, Xi and Yi denote the net insurance loss and the stochastic discount factor of an insurer, respectively. Assume that {(Xi, Yi), i≥1) form a sequence of independent and identically distributed random vectors following a common bivariate Sarmanov distribution. In the presence of heavy-tailed net insurance losses, an asymptotic formula is derived for the finite-time ruin probability.展开更多
Consider an insurance risk model, in which the surplus process satisfies a recursive equationU n =U n?1(1+r n )?X n forn≥1, whereU 0=x≥0 is the initial surplus, {r n ;n≥1} the interest rate sequence, {X n ;n≥1} th...Consider an insurance risk model, in which the surplus process satisfies a recursive equationU n =U n?1(1+r n )?X n forn≥1, whereU 0=x≥0 is the initial surplus, {r n ;n≥1} the interest rate sequence, {X n ;n≥1} the sequence of i. i. d. real-valued random variables with common distribution functionF, which denotes the gross loss during thenth year. We investigate the ruin probability within a finite time horizon and give the asymptotic result asx→∞. Key words variable interest rate - extend regular variation - finite time ruin probability CLC number O 211.9 Foundation item: Supported by the National Natural Science Foundation of China (10071058, 70273029)Biography: WEI Xiao (1979-), female, Ph. D candidate, research direction: large deviations and its applications, insurance mathematics.展开更多
The insurance industry typically exploits ruin theory on collected data to gain more profits.However,state-of-art approaches fail to consider the dependency of the intensity of claim numbers,resulting in the loss of a...The insurance industry typically exploits ruin theory on collected data to gain more profits.However,state-of-art approaches fail to consider the dependency of the intensity of claim numbers,resulting in the loss of accuracy.In this work,we establish a new risk model based on traditional AR(1)time series,and propose a fine-gained insurance model which has a dependent data structure.We leverage Newton iteration method to figure out the adjustment coefficient and evaluate the exponential upper bound of the ruin probability.We claim that our model significantly improves the precision of insurance model and explores an interesting direction for future research.展开更多
In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model....In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model. This paper focuses on the studying of the ruin problems in the above compounded process. In this compounded risk model, ruin may be caused by a claim or oscillation. We decompose the ruin probability for the compounded risk process into two probabilities: the probability that ruin caused by a claim and the probability that ruin caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When the claim sizes are exponentially distributed, the above-mentioned integro-differential equations can be reduced into a three-order partial differential equation.展开更多
Under constant interest force, the risk processes for old and new insurance business are modelled by Brownian motion with drift. By the stochastic control method, the explicit expressions for the minimum ruin probabil...Under constant interest force, the risk processes for old and new insurance business are modelled by Brownian motion with drift. By the stochastic control method, the explicit expressions for the minimum ruin probability and the corresponding optimal strategy are derived. Numerical example shows that the minimum probability of ruin and the optimal proportion for new business decrease as the interest rate increases, and vice versa.展开更多
Consider a nonstandard continuous-time bidimensional risk model with constant force of interest,in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair...Consider a nonstandard continuous-time bidimensional risk model with constant force of interest,in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair of the claim-inter-arrival times is arbitrarily dependent.Under some mild conditions,we achieve a locally uniform approximation of the finite-time ruin probability for all time horizon within a finite interval.If we further assume that each pair of the claim-inter-arrival times is negative quadrant dependent and the two classes of claims are consistently-varying-tailed,it shows that the above obtained approximation is also globally uniform for all time horizon within an infinite interval.展开更多
This paper analyzes a continuous time risk model with a linear model used to model the claim process. The time is discretized stochastically using the times when claims occur, using Doob’s stopping time theorem and...This paper analyzes a continuous time risk model with a linear model used to model the claim process. The time is discretized stochastically using the times when claims occur, using Doob’s stopping time theorem and martingale inequalities to obtain expressions for the ruin probability as well as both expo- nential and non-exponential upper bounds for the ruin probability for an infinite time horizon. Numerical re- sults are included to illustrate the accuracy of the non-exponential bound.展开更多
The paper gives estimates for the finite-time ruin probability with insurance and financial risks. When the distribution of the insurance risk belongs to the class L(γ) for some γ〉0 or the subexponential distribu...The paper gives estimates for the finite-time ruin probability with insurance and financial risks. When the distribution of the insurance risk belongs to the class L(γ) for some γ〉0 or the subexponential distribution class, we abtain some asymptotic equivalent relationships for the finite-time ruin probability, respectively. When the distribution of the insurance risk belongs to the dominated varying-tailed distribution class, we obtain asymptotic upper bound and lower bound for the finite-time ruin probability, where for the asymptotic upper bound, we completely get rid of the restriction of mutual independence on insurance risks, and for the lower bound, we only need the insurance risks to have a weak positive association structure. The obtained results extend and improve some existing results.展开更多
In this paper, we consider the finite time ruin probability for the jump-diffusion Poisson process. Under the assurnptions that the claimsizes are subexponentially distributed and that the interest force is constant, ...In this paper, we consider the finite time ruin probability for the jump-diffusion Poisson process. Under the assurnptions that the claimsizes are subexponentially distributed and that the interest force is constant, we obtain an asymptotic formula for the finite-time ruin probability. The results we obtain extends the corresponding results of Kliippelberg and Stadtmüller and Tang.展开更多
The authors consider two discrete-time insurance risk models. Two moving average risk models are introduced to model the surplus process, and the probabilities of ruin are examined in models with a constant interest f...The authors consider two discrete-time insurance risk models. Two moving average risk models are introduced to model the surplus process, and the probabilities of ruin are examined in models with a constant interest force. Exponential bounds for ruin probabilities of an infinite time horizon are derived by the martingale method.展开更多
This note complements a recent study in ruin theory with risky investment byestablishing the same asymptotic estimate for the finite time ruin probability under a weakerrestriction on the financial risks. In particula...This note complements a recent study in ruin theory with risky investment byestablishing the same asymptotic estimate for the finite time ruin probability under a weakerrestriction on the financial risks. In particular, our result applies to a critical case that theinsurance and financial risks have Pareto-type tails with the same regular index.展开更多
Subject to the assumption that the common distribution of claim sizes belongs to the extendedregular variation class,the present work obtains a simple asymptotic formula for the ruin probability within arandom or nonr...Subject to the assumption that the common distribution of claim sizes belongs to the extendedregular variation class,the present work obtains a simple asymptotic formula for the ruin probability within arandom or nonrandom horizon in the renewal model.展开更多
We consider the Sparre Andersen model modified by the inclusion of interest on the surplus. Approximation for the ultimate ruin probability is derived by rounding. And upper bound and lower bound are also derived by r...We consider the Sparre Andersen model modified by the inclusion of interest on the surplus. Approximation for the ultimate ruin probability is derived by rounding. And upper bound and lower bound are also derived by rounding-down and rounding-up respectively. According to the upper bound and lower bound, we can easily obtain the error estimation of the approximation. Applications of the results to the compound Poisson model are given.展开更多
Consider a discrete-time risk model with insurance and financial risks in a stochastic economic environment. Assume that the insurance and financial risks form a sequence of independent and identically distributed ran...Consider a discrete-time risk model with insurance and financial risks in a stochastic economic environment. Assume that the insurance and financial risks form a sequence of independent and identically distributed random vectors with a generic random vector following a wide type of dependence structure. An asymptotic formula for the finite-time ruin probability with subexponential insurance risks is derived. In doing so, the subexponentiality of the product of two dependent random variables is investigated simultaneously.展开更多
基金supported by the Nature Science Foundation of Hebei Province(A2014202202)supported by the Nature Science Foundation of China(11471218)
文摘This article deals with the problem of minimizing ruin probability under optimal control for the continuous-time compound binomial model with investment. The jump mechanism in our article is different from that of Liu et al [4]. Comparing with [4], the introduction of the investment, and hence, the additional Brownian motion term, makes the problem technically challenging. To overcome this technical difficulty, the theory of change of measure is used and an exponential martingale is obtained by virtue of the extended generator. The ruin probability is minimized through maximizing adjustment coefficient in the sense of Lundberg bounds. At the same time, the optimal investment strategy is obtained.
文摘We consider a risk model with a premium rate which varies with the level of free reserves. In this model, the occurrence of claims is described by a Cox process with Markov intensity process, and the influence of stochastic factors is considered by adding a diffusion process. The integro-differential equation for the ruin probability is derived by a infinitesimal method. Key words ruin probability - variable premium rate - diffusion process - Markov intensity CLC number O 211.9 Foundation item: Supported by the National Natural Science Foundation of China (10071058, 70273029)
文摘In this paper, a new risk model is studied in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is p-thinning process. The integral representations of the survival probability are gotten. The explicit formula of the survival probability on the infinite interval is obtained in the special casc cxponential distribution.The Lundberg inequality and the common formula of the ruin probability are gotten in terms of some techniques from martingale theory.
基金Supported by the Foundation of Suzhou Science and Technology University
文摘This paper studies a Sparre Andersen negative risk sums model in which the distribution of "interclaim" time is that of a sum of n independent exponential random variables. Thus, the Erlang(n) model is a special case. On this basis the correlated negative risk sums process with the common Erlang process is considered. Integro-differential equations with boundary conditions for ψ(u) are given. For some special cases a closed-form expression for ψ(u) is derived.
基金Supported by the National Natural Science Foundation of China (70273029)
文摘Let R(t)=u+ct-∑ I=1^N(t) Xi,t≥0 be the renewal risk model, with Fx(x)being the distribution function of the claim amount X. Let ψ(u) be the ruin probability with initial surplus u. Under the condition of Fx(x) ∈ S^*(γ),y ≥ 0, by the geometric sum method, we derive the local asymptotic behavior for ψ(u,u + z] for every 0 ( z ( oo, On one hand, the asymptotic behavior of ψ(u) can be derived from the result obtained. On the other hand, the result of this paper can be applied to the insurance risk management of an insurance company.
基金Supported by the National Natural Science Foundation of China(11301481,11201422,11371321)Zhejiang Provincial Key Research Base for Humanities and Social Science Research(Statistics)Foundation for Young Talents of ZJGSU(1020XJ1314019)
文摘Consider a continuous-time renewal risk model, in which every main claim induces a delayed by-claim. Assume that the main claim sizes and the inter-arrival times form a sequence of identically distributed random pairs, with each pair obeying a dependence structure, and so do the by-claim sizes and the delay times. Supposing that the main claim sizes with by-claim sizes form a sequence of dependent random variables with dominatedly varying tails, asymptotic estimates for the ruin probability of the surplus process are investigated, by establishing a weakly asymptotic formula, as the initial surplus tends to infinity.
基金Supported by the National Natural Science Foundation of China(11001052,11171065,11326175)China Postdoctoral Science Foundation(2012M520964)+2 种基金Natural Science Foundation of Jiangsu Province ofChina(BK20131339)Postdoctoral Research Program of Jiangsu Province(1302015C)Qing Lan Project and Project of Construction for Superior Subjects of Statistics&Audit Science and Technology of Jiangsu Higher Education Institutions
文摘Consider a discrete-time insurance risk model. Within period i, i≥ 1, Xi and Yi denote the net insurance loss and the stochastic discount factor of an insurer, respectively. Assume that {(Xi, Yi), i≥1) form a sequence of independent and identically distributed random vectors following a common bivariate Sarmanov distribution. In the presence of heavy-tailed net insurance losses, an asymptotic formula is derived for the finite-time ruin probability.
文摘Consider an insurance risk model, in which the surplus process satisfies a recursive equationU n =U n?1(1+r n )?X n forn≥1, whereU 0=x≥0 is the initial surplus, {r n ;n≥1} the interest rate sequence, {X n ;n≥1} the sequence of i. i. d. real-valued random variables with common distribution functionF, which denotes the gross loss during thenth year. We investigate the ruin probability within a finite time horizon and give the asymptotic result asx→∞. Key words variable interest rate - extend regular variation - finite time ruin probability CLC number O 211.9 Foundation item: Supported by the National Natural Science Foundation of China (10071058, 70273029)Biography: WEI Xiao (1979-), female, Ph. D candidate, research direction: large deviations and its applications, insurance mathematics.
基金the Natural Science Foundation of Jilin Province(No.20180101216JC)the National Natural Science Foundation of China(No.11871028).
文摘The insurance industry typically exploits ruin theory on collected data to gain more profits.However,state-of-art approaches fail to consider the dependency of the intensity of claim numbers,resulting in the loss of accuracy.In this work,we establish a new risk model based on traditional AR(1)time series,and propose a fine-gained insurance model which has a dependent data structure.We leverage Newton iteration method to figure out the adjustment coefficient and evaluate the exponential upper bound of the ruin probability.We claim that our model significantly improves the precision of insurance model and explores an interesting direction for future research.
基金The NNSF(10671072,10726075)of Chinathe Doctoral Program Foundation(20060269016)of the Ministry of Education of Chinathe National Basic Research Program(973 Program,2007CB814904)of China.
文摘In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model. This paper focuses on the studying of the ruin problems in the above compounded process. In this compounded risk model, ruin may be caused by a claim or oscillation. We decompose the ruin probability for the compounded risk process into two probabilities: the probability that ruin caused by a claim and the probability that ruin caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When the claim sizes are exponentially distributed, the above-mentioned integro-differential equations can be reduced into a three-order partial differential equation.
基金The National Natural Science Foundations of China (No.10301011 and No.70271069)
文摘Under constant interest force, the risk processes for old and new insurance business are modelled by Brownian motion with drift. By the stochastic control method, the explicit expressions for the minimum ruin probability and the corresponding optimal strategy are derived. Numerical example shows that the minimum probability of ruin and the optimal proportion for new business decrease as the interest rate increases, and vice versa.
基金Supported by the Natural Science Foundation of China(12071487,11671404)the Natural Science Foundation of Anhui Province(2208085MA06)+1 种基金the Provincial Natural Science Research Project of Anhui Colleges(KJ2021A0049,KJ2021A0060)Hunan Provincial Innovation Foundation for Postgraduate(CX20200146)。
文摘Consider a nonstandard continuous-time bidimensional risk model with constant force of interest,in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair of the claim-inter-arrival times is arbitrarily dependent.Under some mild conditions,we achieve a locally uniform approximation of the finite-time ruin probability for all time horizon within a finite interval.If we further assume that each pair of the claim-inter-arrival times is negative quadrant dependent and the two classes of claims are consistently-varying-tailed,it shows that the above obtained approximation is also globally uniform for all time horizon within an infinite interval.
基金Supported by the National Natural Science Foundation of China (Nos. 19831020 and 70003002) and the Fundamental Research Foundation of School of Economics and Management,Tsinghua University
文摘This paper analyzes a continuous time risk model with a linear model used to model the claim process. The time is discretized stochastically using the times when claims occur, using Doob’s stopping time theorem and martingale inequalities to obtain expressions for the ruin probability as well as both expo- nential and non-exponential upper bounds for the ruin probability for an infinite time horizon. Numerical re- sults are included to illustrate the accuracy of the non-exponential bound.
基金Supported by the National Natural Science Foundation of China (No.10671139)
文摘The paper gives estimates for the finite-time ruin probability with insurance and financial risks. When the distribution of the insurance risk belongs to the class L(γ) for some γ〉0 or the subexponential distribution class, we abtain some asymptotic equivalent relationships for the finite-time ruin probability, respectively. When the distribution of the insurance risk belongs to the dominated varying-tailed distribution class, we obtain asymptotic upper bound and lower bound for the finite-time ruin probability, where for the asymptotic upper bound, we completely get rid of the restriction of mutual independence on insurance risks, and for the lower bound, we only need the insurance risks to have a weak positive association structure. The obtained results extend and improve some existing results.
基金Supported by the National Natural Science Foundation of China(No.70471071)Philosophy and Social Science Foundation of the Education Anthority of Jiangsu Province(No.04SJB630005)
文摘In this paper, we consider the finite time ruin probability for the jump-diffusion Poisson process. Under the assurnptions that the claimsizes are subexponentially distributed and that the interest force is constant, we obtain an asymptotic formula for the finite-time ruin probability. The results we obtain extends the corresponding results of Kliippelberg and Stadtmüller and Tang.
基金a grant from National Natural Science Foundation of China,Grant No.10671072Doctoral Program Foundation of the Ministry of Education of China,Grant No.20060269016+1 种基金"Shu Guang"project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation,Grant No.04SG27the National Basic Research Program of China (973 Program),Grant No.2007CB814904
文摘The authors consider two discrete-time insurance risk models. Two moving average risk models are introduced to model the surplus process, and the probabilities of ruin are examined in models with a constant interest force. Exponential bounds for ruin probabilities of an infinite time horizon are derived by the martingale method.
文摘This note complements a recent study in ruin theory with risky investment byestablishing the same asymptotic estimate for the finite time ruin probability under a weakerrestriction on the financial risks. In particular, our result applies to a critical case that theinsurance and financial risks have Pareto-type tails with the same regular index.
基金Supported by the National Statistical Science Research Project (No.LX0317)
文摘Subject to the assumption that the common distribution of claim sizes belongs to the extendedregular variation class,the present work obtains a simple asymptotic formula for the ruin probability within arandom or nonrandom horizon in the renewal model.
基金Supported by the National Natural Science Foundation of China(No.10571051)the Specialized Research Fund for the Doctoral Program of Higher Education of China(20040542006).
文摘We consider the Sparre Andersen model modified by the inclusion of interest on the surplus. Approximation for the ultimate ruin probability is derived by rounding. And upper bound and lower bound are also derived by rounding-down and rounding-up respectively. According to the upper bound and lower bound, we can easily obtain the error estimation of the approximation. Applications of the results to the compound Poisson model are given.
基金Supported in part by the Natural National Science Foundation of China under Grant No.11671012the Natural Science Foundation of Anhui Province under Grant No.1808085MA16the Provincial Natural Science Research Project of Anhui Colleges under Grant No.KJ2017A024 and KJ2017A028
文摘Consider a discrete-time risk model with insurance and financial risks in a stochastic economic environment. Assume that the insurance and financial risks form a sequence of independent and identically distributed random vectors with a generic random vector following a wide type of dependence structure. An asymptotic formula for the finite-time ruin probability with subexponential insurance risks is derived. In doing so, the subexponentiality of the product of two dependent random variables is investigated simultaneously.