Several Risk Models With Parisian Implementation Delay

Recently,the concept of Parisian implementation delay is widely used in risk theory.It is applied in ruin time and dividend payment,respectively in this paper.Two risk models with Parisian implementation delay are discussed.In the first model,the Parisian implementation delay is applied in the ruin issue,that is to say,the ruin will be announced only when the surplus process has continuously stayed below the level 0 for a certain time r(>0).Besides,there is another kind of ruin,the ulti-mate bankruptcy.The ultimate bankruptcy is supposed that when the insurance company's surplus level is negative,the company will continue to operate by filling the temporary d-eficit by means of financing such as bank loans.But the company will declare ruin when the surplus process below the critical level-a(<0).The ultimate bankruptcy is related to the Parisian implementation delay and it extends to a spectrally negative Lévy risk model.This note is dealing with Parisian ruin with an ultimate bankruptcy level-a which is studied in Czarna and Renaud(2016).We extend the spectrally negative Lévy risk process to the refracted Lévy risk process.The probability of Parisian ruin with an ultimate bankruptcy level-a is given by using the strong Markov property and scale functions of the Lévy risk process,.In the second model,the concept of Parisian implementation delay is applied to the dividend payment of risk models,that is to say,the dividend will be paid only when the surplus process has continuously stayed above the barrier b(>0)for a certain time r(>0).Cheung and Wong(2016)have studied the dividend payment with Parisian implementation delay on the dual risk model.We use another method to study the dividend payment with Parisian implementation delay on the dual risk model by scale functions in this paper.And The Laplace transform with scale functions of the ruin time is derived by using the strong Markov property of the risk model.