Asymptotics for the joint tail probability of bidimensional randomly weighted sums with applications to insurance

This paper studies the joint tail behavior of two randomly weighted sums∑_(i=1)^(m)Θ_(i)X_(i)and∑_(j=1)^(n)θ_(j)Y_(j)for some m,n∈N∪{∞},in which the primary random variables{X_(i);i∈N}and{Y_(i);i∈N},respectively,are real-valued,dependent and heavy-tailed,while the random weights{Θi,θi;i∈N}are nonnegative and arbitrarily dependent,but the three sequences{X_(i);i∈N},{Y_(i);i∈N}and{Θ_(i),θ_(i);i∈N}are mutually independent.Under two types of weak dependence assumptions on the heavy-tailed primary random variables and some mild moment conditions on the random weights,we establish some(uniformly)asymptotic formulas for the joint tail probability of the two randomly weighted sums,expressing the insensitivity with respect to the underlying weak dependence structures.As applications,we consider both discrete-time and continuous-time insurance risk models,and obtain some asymptotic results for ruin probabilities.

Markov调节中基于时滞和相依风险模型的最优再保险与投资

本文研究保险公司在Markov调节下基于时滞及相依风险模型的最优再保险与最优投资问题,其中市场被划分为有限个状态,一些重要的参数随着市场状态的转换而变化.假设保险公司的盈余过程由复合Poisson过程描述,而风险资产的价格过程由几何跳扩散模型刻画,并且假设这两个跳过程是相依的.以最大化终端财富值的均值-方差效用为目标,在博弈论框架下,利用随机控制理论和相应的广义Hamilton-Jacobi-Bellman(HJB)方程,本文得到最优策略和值函数的显式表达,并证明解的存在性和唯一性.最后,通过一些数值实例,验证所得结论的正确性,并探讨一些重要参数对最优策略的影响.

Optimal Investment and Premium Control in a Nonlinear Diffusion Model

This paper considers the optimal investment and premium control problem in a diffusion approxi- mation to a non-homogeneous compound Poisson process. In the nonlinear diffusion model, it is assumed that there is an unspecified monotone function describing the relationship between the safety loading of premium and the time-varying claim arrival rate. Hence, in addition to the investment control, the premium rate can be served as a control variable in the optimization problem. Specifically, the problem is investigated in two cases： （i） maximizing the expected utility of terminal wealth, and （ii） minimizing the probability of ruin respectively. In both cases, some properties of the value functions are derived, and closed-form expressions for the optimal policies and the value functions are obtained. The results show that the optimal investment policy and the optimal premium control policy are dependent on each other. Most interestingly, as an example, we show that the nonlinear diffusion model reduces to a diffusion model with a quadratic drift coefficient when the function associated with the premium rate and the claim arrival rate takes a special form. This example shows that the model of study represents a class of nonlinear stochastic control risk model.

Alternative Approach to the Optimality of the Threshold Strategy for Spectrally Negative Lvy Processes

Consider the optimal dividend problem for an insurance company whose uncontrolled surplus precess evolves as a spectrally negative Levy process. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. In this paper, we show that a threshold strategy （also called refraction strategy） forms an optimal strategy under the condition that the Levy measure has a completely monotone density.

Asymptotic Results for Tail Probabilities of Sums of Dependent and Heavy-Tailed Random Variables

Abstract Let X1, X2,... be a sequence of dependent and heavy-tailed random variables with distributions F1, F2,.. on （-∞,∞）, and let T be a nonnegative integer-valued random variable independent of the sequence {Xk, k 〉 1}. In this framework, the asymptotic behavior of the tail probabilities of the quantities Sn = fi Xk and S（n） =∑ k=1 n 〉 1, and their randomized versions ST and S（τ） are studied. Some risk theory are presented. max Sk for 1〈k〈n applications to the

Uniform Asymptotics for Finite-time Ruin Probability in a Dependent Risk Model with General Stochastic Investment Return Process

In this paper,we consider a non-standard renewal risk model with dependent claim sizes,where an insurance company is allowed to invest his/her wealth in financial assets,leading to some stochastic investment log-returns described as a general adapted càdlàg process.Under the assumptions that the claim sizes are heavy-tailed and the stochastic log-return process on investments is bounded from below almost surely,we derive some asymptotic formulas for the finite-time ruin probability holding uniformly in any finite time horizon.

Valuation of CDS counterparty risk under a reduced-form model with regime-switching shot noise default intensities

We study the counterparty risk for a credit default swap （CDS） in a regime-switching market driven by an underlying continuous-time Markov chain. We model the default dependence via some correlated Cox processes with regime-switching shot noise intensities containing common shock. Under the proposed model, the general bilateral counterparty risk pricing formula for CDS contracts with the possibility of joint defaults is presented. Based on some expressions for the conditional Laplace transform of the integrated intensity processes, semi-analytical solution for the bilateral credit valuation adjustment （CVA） is derived. When the model parameters satisfy some conditions, explicit formula for the bilateral CVA at time 0 is also given.