Ruin Probability for Some Mixed Linear Exponential Family in Classical Risk Process

Abstract

This article presents the probability of ruin for the classical risk process by including the density function of claims which satisfies a mixed linear exponential family. This can be defined as <img src=image/13428362_01.gif>, where <img src=image/13428362_02.gif>, <img src=image/13428362_03.gif>, <img src=image/13428362_20.gif> is a positive integer with <img src=image/13428362_04.gif> with <img src=image/13428362_05.gif>, <img src=image/13428362_06.gif>, <img src=image/13428362_07.gif>, and <img src=image/13428362_08.gif> is the canonical parameter. The main results show that the ordinary differential equation for the probability of ruin in the general case by using chain rule and mathematical induction technique is given in Theorem 2.2, the ordinary differential equation for some mixed linear exponential family when <img src=image/13428362_09.gif>, <img src=image/13428362_10.gif>, <img src=image/13428362_11.gif>, <img src=image/13428362_12.gif>, <img src=image/13428362_13.gif>, <img src=image/13428362_14.gif> is demonstrated in Theorem 2.3, and an explicit solution for the probability of ruin when the mixed linear exponential family satisfies the conditions which are <img src=image/13428362_15.gif>, <img src=image/13428362_16.gif>, <img src=image/13428362_10.gif> with <img src=image/13428362_17.gif>, and <img src=image/13428362_13.gif> is indicated in Theorem 2.4. Finally, we use MATLAB to generate the numerical simulations for the probability of ruin in the risk process that the number of claims is a Poisson process and the density function of claims satisfies a mixed linear exponential family and a gamma distribution under the conditions of Theorem 2.4 with the parameters <img src=image/13428362_18.gif>=1 and <img src=image/13428362_19.gif>=0.2. The numerical results reveal that the relative frequency of the ruin and the ruin probability also satisfy the Lundberg inequality which is the necessary condition for the ruin probability. In addition, the absolute values of its differences are small in order to confirm that the main results are correct.