The Application Of Lévy Process In Risk Theory

It is easy to handle the classical risk model,however,its result is not accurate enough, the model itself also has many limitations and inconsistant with the reality. In this paper,we introduce Lévy process to describe the disturbance of the risk management process. Applied to the uncertainty of the premium income and the claim. Lévy process allows the model for more changes,high volatility can be modeled. This paper mainly use Lévy process and its related theory to research the following questions:Firstly,we extend the distribution of the ruin time,the deficit at ruin,the surplus immediately prior to ruin,and get the ruin event cased by the distribution of the ruin time, the deficit at ruin,the surplus immediately prior to ruin and the last minimum surplus level before ruin in finite time and infinite time, respectively. Although this problem has been studied by many scholars and we have achieved some results,It cannot be directly applied to a wider range of the model as the methods related to the model itself.Secondly, in the point of the stochastic process, many models in the continuous risk theory we discussed is Lévy process with jumps,we can use the results of Lévy process applied to the related ruin issues. when Lévy measure belong to subexponential,we got the asymptotic estimate expression of ruin probability in Lévy process and the asymptotic estimate of the tail state of the supremum.Thirdly,we discuss the scale function with the application in spectrally negative Lévy process,we achieved the expression of one- and two-side exit problem according to the scale function. Then we analyse the depletion problem and the drawdowns problem which happened in spectrally negative Lévy process with scale function.