Parisian Ruin Of Risk Models

Ruin probability and ruin time of risk models are important topics in risk management.In this paper we shall consider more general surplus processes including interest rate,called risk reserve processes with constant force of interest.Moreover,we creatively give two general risk models of Gaussian processes including interest rate,which are useful in financial markets due to their special structure.Another contributions of this paper lie in investigating the general concept of risk theory,Parisian ruin,which can better reflect the fluctuation of financial market than the classical ruin.For the Parisian ruin to occur,the risk reserve process must stay below zero for a continuous time interval of pre-specified length.Obviously,it is very difficult to give the exact formula of such Parisian ruin probability of risk models.In order to solve this problem,we introduce generalized Piterbarg constants to give the asymptotics of ruin probability when the initial reserve tends to infinity.Such asymptotics results are very important in practical applications,since they can guide the company when to prevent the bankrupt.In this paper,we find an approximation of the Parisian ruin probability of the Brownian motion risk model with constant force of interest by the Pickands theorem and show that the Parisian ruin time of this risk process can be approximated by an exponential random variable.Our results are new even for the classical ruin probability and ruin time.Furthermore,we also consider risk models without constant force of interest and compare the approximation result with the former ones.Lastly,we give an approximation of the Parisian ruin probability of an integrated Gaussian risk model with constant force of interest by the Pickands theorem and the method of moments as the initial reserve goes to infinity and the pre-specified length of time goes to zero.Then,we are interested in a large deviation type result for the Parisian ruin probability.Specifically,under two weak restrictions on the process and the inflation/deflation rate function,we shall show in our first result that the Parisian ruin probability and the classical ruin probability are on the log-scale asymptotically the same.Such a result reveals that the asymptotic of Parisian ruin probability and the classical one are the same also in the precise asymptotics behavior.Moreover,we are concerned with the distribution of the Parisian ruin time,since it gives more information on how and when the ruin has occurred.The result reveals that the Parisian ruin time of risk processes can be approximated by an exponential random variable.