The Cumulative Discounted Expected Dividend Value Under The Ratchet Strategy

De Finetti proposed the concept of dividends in 1957.Since then,the study of optimal divdend strategies in risk theory has been a very active field for more than 60 years.The optimal Barrier dividend strategy and the optimal Threshold dividend strategy have been proposed successively.In 1903,the development of Cramer-Lundberg risk model lays an important foundation for the development of risk theory.On this basis,people have further expanded the optimal dividend problem,and increased many financial and insurance issues,such as capital injection and liquidation,etc.Subsequently,the study of Levy process provided new ideas for risk theory.Many scholars have replaced the risk model with the Levy process.Based on that,they have studied the optimal dividend problem and the optimal capital injection problem with new methods.In 2018,a new dividend strategy,Ratchet dividend strategy,was mentioned by Albrech-er.This new dividend strategy regulates that once the dividend rate rises to a higher level,it will no longer decline.It makes the whole revised earnings process no longer Markov,which is very novel.Based on the spectrally negative Levy process,the Ratchet dividend strategy is studied by Albrecher.In this paper,we use the scale function and the fluctu-ation theory of Levy process.We give the cumulative discounted expectation function of n-weighted Ratchet dividend strategy,2-weighted Ratchet dividend strategy and the cumu-lative discounted expectation function of Ratchet dividend strategy under the positive Levy process.The structure of this paper is as follows:Chapter one is the introduction.This part mainly introduces the research history and recent situation of risk theory and dividend problems,and finally introduces Ratchet dividend strategy.The second chapter is the introduction of the model.In this part,the Ratchet dividend strategy and the model to be studied are introduced in detail,and the value function to be studied is established.Chapter three is divided into two parts.The first part mainly introduces the spectrally negative Levy process,scale function and the fluctuation theory under the spectrally negative Levy process.In thesecond part,the main conclusions are given,the cumulative discounted expectation function of n-weighted Ratchet dividend strategy and the expectated time of ruin under 2-weighted Ratchet dividend strategy are obtained.The detailed proof and calculation process are given.At the same time,some common examples aregiven.Chapter four is also divided into two parts.The first part mainly introduces the spec-trally positive Levy process,scale function and the fluctuation theory under the spectrally positive Levy process.In the second part,we give the cumulative discounted expectation function of Ratchet dividend strategy under the spectral positive Levy process and an ex-ample of dual model.Finally,the fifth chapter is a summary and prospect.