The Martingale Property Of Local Martingales And Exponential Local Martingales

Girsanov theorem is an important theorem in the area of financial mathematics.When using the theorem,we always need to judge whether a exponential local martingale is an uniformly integrable martingale or not.On the other hand,because any exponential local martingales is a local martingale,judging whether a local martingale is an uniformly integrable martingale or not is also a problem worth discussing.This dissertation is mainly a paper of literature reviews.It will introduce the main conclusions made by other researchers about the two questions we mentioned,mainly including the conclusions given by Novikov(1973),Kazamaki(1977),Krylov(2002),Lepingle et al.(1978),Kazamaki et al.(1983),Rao(1969)and Hulley et al.(2019).According to exponential martingales,based on the Novikov criterion and Kazamaki criterion,researchers presented a series of sufficient conditions to deal with more situations.According to local martingales,researchers presents some sufficient conditions and equivalent conditions under different restrictions of the local martingales.The passage will choose some of the conclusions to introduce.In this dissertation,the first chapter briefly introduces the results of the questions,and shows the structure of the paper.The second chapter gives the necessary definitions and lemmas that are necessary for the following parts,and discusses the background of the problem to show its importance.The third chapter discusses how to judge whether an exponential martingale is an uniformly integrable martingale or not,shows the main results given by other researchers,and introduces some examples to show the application of the results.The fourth chapter discusses how to judge whether a local martingale is an uniformly integrable martingale or not,shows the main results given by other researchers,introduces some related examples,and tries to use the results to reanalyse the problem in the third chapter.