Martingale Transform And Related Problems

The present Ph.D. thesis mainly deals with the the martingale transform for right processes and related problems. Firstly, we consider the problem of the representation formula for the transition probability density of a right process under Girsanov transform. Secondly,we consider the problem of the MEMMs (minimal entropy martingale measure)for Markov switching Levy processes, and justify that the minimal entropy martingale measure is obtained by some regime switching Esscher transform. At last,we consider the problem of the invariant measure and the ergodic property of recurrent right processes.This paper includes 5 chapters. In chapter 1, we introduce the background of the third,the fourth and the fifth chapters and our results. In chapter 2, we introduce some important concepts,theorems and properties which are used in the following three chapters. For the content of the second chapter, we can see [1],[15], [16], [25],[26],[31].In chapter 3, we obtain the representation formula for the transition probability density of a right process under Girsanov transform. Qian and Zheng ([24]) established a representation formula for the transition probability density of a diffusion perturbed by a vector field. In the case of [24], they considered the diffusion processes where the Levy systems disappeared. In this chapter, we shall consider a much more general case that is a right process with jumps.While in our case, since the process has jumps, we have to compute Levy systems of the Markovian bridge (which is firstly defined in [11]) and the transformed process. The representation formula for the transition probability density of a right process under Girsanov transform we obtained is very useful in obtaining information about the density functions perturbed by a drift transform. Therefore it has theoretic and practical values by its own. We also obtain the representation formula for the transition probability density of a right process under Esscher transform and the infinitesimal generator of the transformed process.In chapter 4,we obtain the MEMMs for the Markov switching Levy processes. The MEMMs for Levy processes and geometric Levy processes have been studied by [9]and [12]. Elliott,Chan and Siu ([6]) investigated the option pricing problem when the risky underlying assets were driven by Markov-modulated Geometric Brownian Motion. There they adopted the regime switching Esscher transform which was the modification of the random Esscher transform introduced by [28]. They justified their pricing result by the minimal entropy martingale measure. Elliott and Osakwe ([8])studied the option pricing for pure jump processes with Markov switching compensators. In this chapter,we investigate the option pricing problem when the risky underlying assets are driven by Markov-modulated Levy process and we justify that the minimal entropy martingale measure is obtained by some regime switching Esscher transform.In chapter 5, we obtain the invariant measure and the ergodic property of recurrent right processes. For a positive recurrent Markov chain, there exists a unique invariant distribution. It is interesting to study that under what conditions a Markov process has an invariant measure. This question has been studied by many authors. [21] and [29] had proved the existence and uniqueness of an invariant measure and the ergodic property for one-dimentional recurrent diffusion processes. Khas'minskii ([19]) studied the ergodic properties of recurrent diffusion processes on aσ-compact complete metric space. Maruyama and Tanaka ([22])studied the same questions for recurrent Markov process in N-dimentional Euclidean space R~N which have right continuous paths and the strong Markov property. In this chapter, we consider the recurrent right process on Polish space.