Research On The Bankruptcy Problem Of Discrete Time Dependent Risk Model Under Markov Chain Rate

With the diversification of economic,the introduction of interest rate into risk model is a research topic which is wildly concerned by the modern risk theory,and it enhances the realistic description of the model.That is why the discrete risk model based on Markov chain with interest rate receives extensive attention in actuarial literature.This paper derives the lower bound estimation of the bankruptcy probability and other characteristics based on the discrete time dependent risk model.We assume that the interest rate is described by Markov chain,the premium and claim processes are two different high order autoregressive structures.The high-order autoregressive structure is more representative than the first-order autoregressive structure in time series analysis.That is,in real life,the premium and claim amount are not only related to the previous stage,but may be related to the previous n stages.In addition,the higher order autoregressive structure possess several values,and it may bring difficulties in proof and calculation.Firstly,with the help of renewal recursion,we gain the integral equation that satisfying the ruin probability and the lower bound estimation of the ruin probability.Secondly,the recursive formula of the maximum surplus distribution before bankruptcy is obtained.Based on those results,the integral equation of the joint distribution of the surplus before bankruptcy,the deficit after bankruptcy and the maximum surplus before bankruptcy is derived.Finally,we gain the recursive formula of the distribution of bankruptcy duration and the time distribution of the surplus for the first time through a certain level.