Constant barrier strategies in a two-state Markov-modulated dual risk model

In this paper,we consider the dividend problem in a two-state Markov-modulated dual risk model,in which the gain arrivals,gain sizes and expenses are influenced by a Markov process.A system of integrodifferential equations for the expected value of the discounted dividends until ruin is derived.In the case of exponential gain sizes,the equations are solved and the best barrier is obtained via numerical example.Finally,using numerical example,we compare the best barrier and the expected discounted dividends in the two-state Markov-modulated dual risk model with those in an associated averaged compound Poisson risk model.Numerical results suggest that one could use the results of the associated averaged compound Poisson risk model to approximate those for the two-state Markov-modulated dual risk model.     

Precise large deviations for sums of random vectors in a multidimensional size-dependent renewal risk model

Consider a multidimensional renewal risk model, in which the claim sizes {X_k, k ≥1} form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Suppose that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. A precise large deviation for the multidimensional renewal risk model is obtained.     

A Class of Ruin Probability Mo del with Dep endent Structure

In this paper, we study a class of ruin problems, in which premiums and claims are dependent. Under the assumption that premium income is a stochastic process, we raise the model that premiums and claims are dependent, give its numerical characteristics and the ruin probability of the individual risk model in the surplus process. In addition, we promote the number of insurance policies to a Poisson process with parameter λ, using martingale methods to obtain the upper bound of the ultimate ruin probability.     

Ruin probabilities for a risk model with two classes of claims

In this paper we consider a risk model with two kinds of claims, whose claims number processes are Poisson process and ordinary renewal process respectively. For this model, the surplus process is not Markovian, however, it can be Markovianized by introducing a supplementary process, We prove the Markov property of the related vector processes. Because such obtained processes belong to the class of the so-called piecewise-deterministic Markov process, the extended infinitesimal generator is derived, exponential martingale for the risk process is studied. The exponential bound of ruin probability in iafinite time horizon is obtained.     

First passage models for denumerable semi-Markov decision processes with nonnegative discounted costs

This paper considers a first passage model for discounted semi-Markov decision processes with denumerable states and nonnegative costs.The criterion to be optimized is the expected discounted cost incurred during a first passage time to a given target set.We first construct a semi-Markov decision process under a given semi-Markov decision kernel and a policy.Then,we prove that the value function satisfies the optimality equation and there exists an optimal(or e-optimal) stationary policy under suitable conditions by using a minimum nonnegative solution approach.Further we give some properties of optimal policies.In addition,a value iteration algorithm for computing the value function and optimal policies is developed and an example is given.Finally,it is showed that our model is an extension of the first passage models for both discrete-time and continuous-time Markov decision processes.     

Option pricing when the regime-switching risk is priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.     

Smoothness of certain functions in two kinds of risk models with a barrier dividend strategy

In this paper,we study the smoothness of certain functions in two kinds of risk models with a barrier dividend strategy.Mainly using technique from the piecewise deterministic Markov processes theory,we prove that the function is continuously differentiable in the first risk model.Using the weak infinitesimal generator method of Markov processes,we prove that the function is twice continuously differentiable in the second risk model.Intego-differential equations satisfied by them are derived.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

RUIN PROBABILITY IN A SEMI-MARKOV RISK MODEL WITH CONSTANT INTEREST FORCE AND HEAVY-TAILED CLAIMS

In the present paper, we consider a kind of semi-Markov risk model （SMRM） with constant interest force and heavy-tailed claims~ in which the claim rates and sizes are conditionally independent, both fluctuating according to the state of the risk business. First, we derive a matrix integro-differential equation satisfied by the survival probabilities. Second, we analyze the asymptotic behaviors of ruin probabilities in a two-state SMRM with special claim amounts. It is shown that the asymptotic behaviors of ruin probabilities depend only on the state 2 with heavy-tailed claim amounts, not on the state 1 with exponential claim sizes.     

马尔可夫交换Lévy过程模型下的期权定价及其对冲（英文）

In this paper, we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging. In this model, the market interest rate, the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process. We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure. The option price using this model is obtained by the Fourier transform method. We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

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??This paper considers the expected penalty functions for a discrete semi-Markov risk model, which includes several existing risk models such as the compound binomial model (with time-correlated claims) and the compound Markov binomial model (with time-correlated claims) as special cases. Recursive formulae and the initial values for the discounted free penalty functions are derived in the two-state model by an easy method. We also give some applications of our results.     

常数红利边界下的一类马氏风险模型

本文研究了常数红利边界下一类马氏风险模型的红利派发矩,破产前所有红利的分布等相关问题.利用更新方法,给出了该模型破产前红利折现的期望满足的微分-积分方程,得到破产前所有红利的分布.通过构造特殊的初始条件,得到了相关的方程组解,推广了文献[3]的结果.     

Entropy Maximization for Markov and Semi-Markov Processes

The literature about maximum of entropy for Markov processes deals mainly with discrete-time Markov chains. Very few papers dealing with continuous-time jump Markov processes exist and none dealing with semi-Markov processes. It is the aim of this paper to contribute to fill this lack. We recall the basics concerning entropy for Markov and semi-Markov processes and we study several problems to give an overview of the possible directions of use of maximum entropy in connection with these processes. Numeric illustrations are presented, in particular in application to reliability.     

Strong Law of Large Numbers and Central Limit Theorems for Functionals of Inhomogeneous Semi-Markov Processes

Limit theorems for functionals of classical (homogeneous) Markov renewal and semi-Markov processes have been known for a long time, since the pioneering work of Pyke Schaufele (Limit theorems for Markov renewal processes, Ann. Math. Statist., 35(4):1746–1764, 1964). Since then, these processes, as well as their time-inhomogeneous generalizations, have found many applications, for example, in finance and insurance. Unfortunately, no limit theorems have been obtained for functionals of inhomogeneous Markov renewal and semi-Markov processes as of today, to the best of the authors’ knowledge. In this article, we provide strong law of large numbers and central limit theorem results for such processes. In particular, we make an important connection of our results with the theory of ergodicity of inhomogeneous Markov chains. Finally, we provide an application to risk processes used in insurance by considering a inhomogeneous semi-Markov version of the well-known continuous-time Markov chain model, widely used in the literature.     

A characterization for mixtures of semi-Markov processes

Mixtures of recurrent semi-Markov processes are characterized through a partial exchangeability condition of the array of successor states and holding times. A stronger invariance condition on the joint law of successor states and holding times leads to mixtures of Markov laws.     

Ruin Probabilities in Cox Risk Models with Two Dependent Classes of Business

In this paper we consider risk processes with two classes of business in which the two claim-number processes are dependent Cox processes. We first assume that the two claim-number processes have a two-dimensional Markovian intensity. Under this assumption, we not only study the sum of the two individual risk processes but also investigate the two-dimensional risk process formed by considering the two individual processes separately. For each of the two risk processes we derive an expression for the ruin probability, and then construct an upper bound for the ruin probability. We next assume that the intensity of the two claim-number processes follows a Markov chain. In this case, we examine the ruin probability of the sum of the two individual risk processes. Specifically, a differential system for the ruin probability is derived and numerical results are obtained for exponential claim sizes.     

马氏风险过程

研究了一般马氏风险过程,它是经典风险过程的拓广．具有大额索赔的风险过程用此马氏风险模型来描述是适合的．在此模型中,索赔到达过程由一点过程来描述,该点过程是一马氏跳过程从0到t时间段内的跳跃次数．主要研究了此风险模型的破产概率,得到了破产概率满足的积分方程,并应用本文引入的广更新方法,得到了破产概率的收敛速度上界．     

考虑Markov保费和利率的离散时间风险模型的破产概率

研究一类离散时间风险模型的破产概率.在保费收入和利率同时为离散时间Markov链,索赔额为独立情形下,利用更新迭代方法得到最终时间破产概率的Lundberg型上界.     

A Markov Risk Model with Two Classes of Insurance Business

A Markov risk model with two classes of insurance business is studied. In this model, the two classes of insurance business are independent. Each of the two independent claim number processes is the number of jumps of a Markov jump process from time 0 to t, whichever has not independent increments in general. An integral equation satisfied by the ruin probability is obtained and the bounds for the convergence rate of the ruin probability are given by using a generalized renewal technique.     

证券组合的Markov链模型

本文将证券价格时间序列分解成趋势变动序列和 Markov链 ,建立了证券组合的 Markov链模型 ,应用 Markov链理论对此模型进行了分析 ,给出了充分大的一个时间内的收益率 ,风险和切点组合的计算公式