Study On Numerical Solution Of Insurance Ruin Probability

Insurance refers to the behavior that the applicant pays the insurance premium to the insurer according to the contract,and the insurer is responsible for compensating the property loss caused by the possible accident agreed in the contract,or paying the insurance benefit when the insured dies,is disabled and reaches the age and time limit agreed in the contract.Insurance companies bear the risks of policyholders,and the risks of insurance companies need to be predicted by insurance risk models.Insurance risk model is a quantitative analysis and prediction of the risk of insurance companies,which studies the impact of variables such as initial capital,premium interest rate,claim arrival process and claim amount on the insurance company risk.The ruin probability in insurance risk model is one of the most concerned variables of insurance companies,however,in most cases,the ruin probability is not easy to solve,which brings inconvenience to insurance companies in managing ruin risk.Based on the classical insurance risk model and the renewal risk model,this thesis studies the numerical solutions of ruin probability under classical insurance risk model and the renewal risk model.This thesis transforms the calculation of ruin probability into an the calculation of integral-differential equation.First,the first and second order precision numerical schemes of ruin probability are obtained by using finite difference method and compound quadrature formula;Secondly,numerical experiments show that the numerical solutions converges and are insensitive to parameter changes.Finally,this thesis compares the results of numerical method and Monte Carlo method in solving ruin probability,numerical experiments show that the numerical method is more accurate than Monte Carlo method and saves a lot of time.The main work and innovations of this thesis are as follows:1.Give the numerical solution of ruin probability under classical risk model.In the classical risk model,the claim arrival process is poisson process,the ruin probability satisfies the first order differential integral equation and has an initial value at zero.We use the finite difference method to approximate the differential and use the compound quadrature formula approximate the integral.Numerical experiments show that the numerical scheme is stable and convergent,and insensitive to parameter variation.2.Give the numerical solution of ruin probability under renewal risk model.In the renewal risk model,the claim arrival process is the renewal process.While claim arrival time interval obey the Erlang(2)distribution,the ruin probability satisfies the second order differential integral equation.For the integration approximation,the compound trapezoidal formula is adopted for the accuracy and simplicity of calculation.For differential approximation,we use two methods,one is to use direct difference method to get the first order precision and second order precision numerical format;The other is to transform the second order differential integral equation into the first order differential integral equation set by variable substitution,and the we can also get the first order precision and second order precision numerical scheme.Numerical experiments show that the numerical scheme is stable and convergent,and insensitive to parameter variation.3.Use Monte Carlo simulation and numerical scheme respectively to approximate solution of ruin probability,the results show that the accuracy of numerical method is higher than that of Monte Carlo simulation,and the operation efficiency of numerical method is much higher than that of Monte Carlo simulation.This thesis provides a reference for solving the ruin probability,the numerical solution of ruin probability is more accurate and time-saving than Monte Carlo random simulation.This thesis provides a reference for insurance companies to grasp the operation status in time and establish financial early warning.