Asymptotics Of The Overshoot And Undershoot Of Random Walk And Lévy Process And Their Applications

In this paper, we will discuss the asymptotics of the overshoot and undershoot of a random walk or a Levy process. Overshoot means a given level is reached at some time. And overshoot can be applied in many fields, such as queueing theory, risk theory, branching process theory and infinite divisible distributions. So, the problem of overshoot is worth to be studied. In risk theory, overshoot has a close relation with the ruin problem, that is, overshoot means' ruin' occurs. So, ruin probability is the probability of the events of overshoot. Ruin probability is an important object, which describe the possibility of the ruin event and measure the risk of an insurance company, Therefore, it has been focused on by many scholars. But ruin probability can't show the severity of the deficit. So, many researchers begin to investigate the asymptotics of moments of the overshoot which reflect some average deficit. On the other hand, undershoot reflects the surplus before ruin. Certainly, the deficit and surplus above have theoretical significance and realistic value for the risk assessment and risk management of insurance companies.In risk theory, there are various risk models to deal with the complicated realistic environment. And these models usually are the extension of the classical renewal model, such as, replace the independent relation among variables by some dependent relation; consider the effect of finance risk as well as the effect of insurance risk. Recently, Levy insurance process has been paid special attention to. This process generalizes some familiar process, for example, a risk process perturbed by Brownian motion, a risk process perturbed by p-stable Levy motion and so on. Since the renewal model can be deduced to a random walk and Levy process plays a major role in a risk process, then we study the the asymptotics of the overshoot of a random walk or a Levy process.Additionally, many phenomena show a big-jump principle plays a major role in insurance industry, that means a large claim often causes the ruin (See Embrechts et al.(1997). So, this paper focuses on the results in the heavy-tailed case. This means we usually assume the distribution of the claim is heavy-tailed, for a Levy process, assume the distribution which is formed by its corresponding Levy measure is heavy-tailed. The most important subfamily of heavy-tailed distribution often refers to subexponential class or local subexponential class. The subexponential class can fully reflect a big-jump principle, and be accord with reality. At the same time, it often can be proved to be equivalent condition of asymptotics of ruin probability as well as sufficient condition. So this assumption is now thought to be quite reasonable.Finally, we point out that since the deficit at ruin can not be infinite, so we pay special attention to the local results. It is well known that when the distribution of claim is light-tailed, its local probability and its nonlocal probability are asymptotically equivalent. Thus, it is not necessary to discuss the local ruin probability. However, when the distribution of claim is heavy-tailed, its local probability often becomes a infinitesimal of its nonlocal probability. In other word, the cost of local risk prevention of a insurance company is less than that of the total risk prevention. Therefore, to some extent, the study of local ruin probability have more realistic value than the study of nonlocal ruin probability.This paper is organized as the following four chapters.In the first chapter, we introduce some notions, notation and concepts which include random walk theory, Levy process theory and some distribution classes.In the second chapter, we obtains some equivalent conditions and sufficient con-ditions for the local and non-local asymptotics of theφ-moments of the overshoot and undershoot of a random walk, whereφis a non-negative and long-tailed function. By the strong Markov property, it can be found that the moments of the overshoot and un-dershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore the paper first investigates the local and non-local asymptotics for the moments of the first ascending ladder height of a random walk, and then gives some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Finally, we get the main results of this chapter.In the third chapter, by the method of path decomposition of Levy process, we obtain the uniform local asymptotics for a Levy process with a heavy-tailed Levy mea-sure. As applications, we get the uniform asymptotics of the finite-time ruin probability for the Levy risk model with a heavy-tailed Levy measure, in other word, we can get asymptotic estimation of the finite-time ruin probability. From the above results, we find that in the compound Poisson model perturbed by a Brownian motion, the of-fect of the Brownian component on the asymptotics of the finite-time ruin probability washes out.In the fourth chapter, we obtain the uniform local asymptotics for the overshoot and undershoot of the Levy process with a heavy-tailed Levy measure. As applica-tions, we get the uniform asymptotics of the local ruin probability for the Levy risk model with a heavy-tailed Levy measure. From the above results, we find that in the compound Poisson model perturbed by a Brownian motion, the effect of the Brownian component on the asymptotics of the local ruin probability washes out. Finally, using the uniform asymptotics of the Levy process, we discuss the asymptotics of moments of its overshoot.