跳跃-扩散模型的首达时研究

A Study about the First Reaching Time in a Jump-Diffusion Risk Model

考虑跳跃-扩散风险模型,研究盈余达到下界L的首达时T的特性.利用更新论证得到关于(u)=E[e-rT|U(0)=u]的更新方程.对于下跳模型,若索赔额为相互独立且具有相同的指数分布,得到更新方程解的解析表示;对于上跳模型,则解析表示的推出不需要指数分布的假定.作为应用,得到了首达时T的均值和方差的表达式.最后给出了数值计算和随机模拟的实例.

In this paper we consider a jump\|diffusion risk model and study the time T that the surplus reaches a lower bound L for the first time.We derive the renewal equation of the function φ(u)=E using the renewal argument. For the jump downward model when the claim amounts are i.i.d. and have an exponential distribution, we obtain an analytical expression for the solution of the renewal equation. For the jump upward model, the assumption about exponential distribution is not needed. As an application, we obtain an expression for the mean and variance of the first reaching time T. For a concrete model numerical and stochastic simulation results are given.