Further Research On The Lévy Risk Models

The Levy process is an extremely important stochastic process in probabilitytheory and statistics,because it has unique properties such as stationary independent increment,Markov property,infinite divisibility,and the sample path of the Levy process has breakpoints,which makes it particularly suitable for simulating sudden changes in financial asset prices.This non-continuous nature enables it to better respond to unexpected events in the financial market and more accurately depict the dynamic changes in financial data.In short,the Levy process,through its unique jumping characteristics,can more accurately simulate the operating rules of the financial market,more effectively depict the asset operation process,and provide powerful mathematical tools for risk management in the financial market.The Lévy process has a wide range of applications as a modeling tool in finance,insurance,and other fields.One of the fundamental issues involves the distribution characteristics of the time when the Levy process first exits an infinite or finite interval,as well as its overshoot and undershoot outside the interval boundary.The integral transforms of the joint distribution of passage time and overshoot/undershoot outside the interval boundary was obtained for the homogeneous process with independent increments by Pecherskii and Rogozin(1969).Based on the joint distribution of one-boundary functionals,Kadankov and Kadankova(2005)determined the Laplace transform for the joint distribution of the passage time and overshoot/undershot.For some special Levy processes,closed form expressions can be obtained,for example,the double exponential jump-diffusion process,the Meromorphic Levy process;the hyper-exponential jump-diffusion process and the compound renewal process,closed form expressions can be obtained.We found that these papers were established for certain specific subclasses of Levy processes,which can only solve certain types of jump distributions and there is not much research on more general Levy processes,this thesis further studies these issues.The Levy process is widely used to model the earnings and claims processes of insurance companies,among which the spectral negative Levy process,as a subclass of the Levy process,only has a downward jump and is commonly used to describe phenomena such as stock price declines or insurance company claims.Compared to classical risk models,the spectral negative Levy process is a type of time continuous process with stationary independent increments and is càdlàg,which makes it more suitable for characterizing the earnings process of insurance companies.The issue of dividends has always been a concern in actuarial science,and how to distribute dividends is a concern for company management and shareholders.In recent years,the ratcheting dividend strategy has received widespread attention,which was proposed by Albrecher et al.(2018).Under this dividend strategy,the dividend rate does not decrease over time.Once a company’s earnings process reaches a given barrier,the dividend rate will increase and remain high until ruin.Generally speaking,different dividend strategies are studied separately,and in recent years,more realistic mixed dividend strategies have attracted many scholars to study.But we found that a mixed dividend strategy combining ratchet dividend strategy and obstacle dividend strategy still needs further research.This thesis considers the one-sided passage problems from below or above and the two-sided exit problems from a finite interval.Firstly,we consider that the Lévy process has a characteristic exponent in which one of the two jumps having a rational Laplace transform.We obtain the joint distribution of first passage time and Overshoot/Undershoot.Next,we consider the Levy process with characteristic exponent and both jumps having rational Laplace transforms.We derive the joint distribution of the first passage time and overshoot/overshoot of the Levy process within a finite interval.This process recover many models that have appeared in the literature such as the compound Poisson risk models,the perturbed compound Poisson risk models,and their dual models.As application,we obtained solutions for popular path-dependent options in terms of Laplace transform,such as lookback options and barrier options.As an application,the dividend problem of the spectral negative Levy risk model is considered,where the dividend strategy is a mixed ratcheting and barrier dividend strategy.Given a barrier level of a<b,when the earnings level does not exceed a,the company distributes dividends at a lower dividend rate.Once the surplus level exceeds a,the dividend rate increases,and all of the excess over b as lump sum dividend payments.Using fluctuation identities and scale functions,we obtain explicit formulas for the expected net present value of dividends until ruin and the Laplace transform of the time to ruin.And numerical illustrations are present to show the impacts of parameters on the expected net present value.