A Bivariate Correlation Risk Model With Mutual Deficit Coverage

Risk is an essential feature of modern finance,and the survival probability of insurance companies is an important part of financial risk theory research.From the point of view of risk management,insurance companies cannot operate in isolation from financial markets or from other insurance and reinsurance companies.Insurance companies and policyholders have to make a choice between the risks and benefits in order to obtain the maximum benefit under certain risk or to guarantee a certain profit under the minimum risk.Relatively speaking,mutual insurance has certain advantages.Such as ship collision,oil pollution,hurricanes and earthquakes and other catastrophic events caused by large-scale claims occurred.Different insurance companies transfer some risks and profits to another insurance company in order to avoid Bankruptcy through mutual insurance.In the process of researching risk theory,a variety of risk models are proposed.Surplus process is the core of risk theory research.Therefore,under the condition of mutual insurance,the research of the surplus problem of two companies,namely research of the ruin probability(survival probability)has a certain theoretical value and practical significance.In this thesis,we first give some theoretical knowledge about the risk model.Then under the bivariate Cramer-Lundberg risk process,and based on the two insurance companies owned mutual agreement,which each insurance company agrees to cover the deficit of the other,we consider three cases.Firstly,this thesis considers two insurance companies claim arrival rate submit to the non-homogeneous Poisson process.An upper bound of finite-time ruin probability is obtained by martingale approach,and then combines with kernel equation of the two-dimensional Laplace transform of survival probability,a lower bound of survival probability under the bivariate Cramer-Lundberg risk process with two insurance companies is obtained.And we acquire a lower bound estimation of the survival probability of the two insurance companies with individual claims submit to exponential distribution.Secondly,we consider dependent risk model about the claim amounts of the two insurance companies,based on Copula while the two companies go bankruptcy simultaneously.The integro-differential equation of the joint survival probability of two insurance companies can be obtained from the bivariate Cramer-Lundberg risk model with claim arrival rate obeys to homogeneous compound Poisson process.Also the recursion formula of the joint survival probability of the two insurance companies can be obtained from the bivariate compound binomial model.Finally,we research the ruin probability of two insurance companies with uncertain income or expenditure,and the expression of the ruin probability is obtained.In the end of the thesis,the results of the research are summarized,and the prospect of the thesis is obtained.