Research On Two Types Of Lévy Risk Models With Parisian Delay

The concept of Parisian ruin originated from Parisian options.There are two major types of Parisian ruin,one is deterministic delays,and the other one is stochastic delays.The Parisian ruin under a hybrid observation scheme was introduced by [25],which includes both deterministic and stochastic delays.Under the hybrid observation scheme,the risk process is first monitored discretely at Poisson arrival epochs until a negative surplus is observed.Then the risk process will be observed continuously in a grace period > 0.If the surplus recover to a solvable level (6 ? 0 in the grace period,then the hybrid observation will return to discrete Poisson observations.Otherwise,Parisian ruin will be declared at the end of the grace period.[25] studied the Parisian ruin problem for spectrally negative Lévy processes under a hybrid observation scheme,and obtained the specific expressions for the probability of Parisian ruin and its limiting cases.On this basis,[28] discussed the Laplace transform of the Parisian ruin time when (6 = 0.In this paper,we discuss two types of risk surplus process with Parisian delays.In the first part,we continue to discuss the Paisian ruin problem based on a hybrid observation scheme for a spectrally negative Lévy process.We first discuss the Laplace transform of the surplus at ruin when (6 ? 0,then we study the joint Laplace transform of the ruin time and the deficit when (6 = 0,and we give the specific expressions by scale functions.In the process of obtaining the joint Laplace transform,we mainly based on the strong Markov property.In addition,the use of Laplace transform makes our proof avoid any discussion on bounded or unbounded variation paths of the surplus process.Actually,hybrid observation scheme can also be applied in dividend problem.Therefore,in the second part,we study the barrier dividend problem under a hybrid observation scheme for a spectrally positive Lévy process.Specifically,when the insurance surplus is above 0,it is monitored discretely at Poisson arrival epochs until the surplus is observed above the barrier (7 > 0,then the process will be observed continuously in a time lag > 0.If the surplus keeps staying above the barrier during the time lag,the excess will be paid at the end of the time period.Otherwise,we are going to return to the discrete observations.We obtain the joint Laplace transform of the time to ruin and the total dividends until ruin time and also give the specific expressions by scale functions.