Martingale With Applications In Finance

The original intention of "martingale" is "being fair to gamble". It is first to learned from French probability expert P. Levy to usher in the begining of 20th century. The year of 1953, after Levy making some initial operates, the American famous probability leaners J. L. Doob introduced to martingale(process)theory in random process firstly, making it becomes an important independent sub-branch of random process theory. These premises are martingale and stochastic integral the theoretic development developed new way and also provided powerful tool for its subsequent.Along with the quick shape of martingale, at present it has become various probability that more has depth in point of it a standard composition of the appli-cation work fraction, and in the random differential equation. the theory and preface be presumed analytical and random control to ect. realm to have an important ap-plication.The begin of the text is an introduction, introduced shape back ground and related progress of martingale theory. The No.2 chapter is the point of this text search, systematically elaborated the related theory of discrete time martingale. The framework that includes the definition of discrete time martingale, submartingale inequality and its decomposition and convergence theory etc. The chapter finally still clearly gave the definition and construction of two process of martingale. In this chapter my main operate has the following points:(1)In originalling possessed the foundation of martingale definition, give con-cretely solid instance within real life, for example, toss specie, share in the stock market etc. enunciation martingale has comprehensive in each realm application.(2)To freeze-out has the restriction of boundary towards stopping times and get conclusion:While stopping times can keep structure of martingale(submartingale).(3)The approval differs the definition of the row to the L2 martingale, cancell the limit system of stopping times, prove the expection of square about sum of martingale is smaller than the expection of sum about the square of martingale.(4)As appliation, combine with the row to martingale, and two process of mar-tingale, then judge its quality of martingale(submartingale).Analogy as the discussion of No.2 chapter, the third chapter discuss continuous time martingale as to it's also the bastic quality, convergence quality and decompo- sition made a detailed present. My main work of this chapter as follows:(1)Compare two process of discrete time martingale to continuous time mar-tingale, give the definition of submartingale orthogonal and judge it.(2)Give and proof the translation condition that continuous time martingale to the process of familiar D.The forth chapter is a pais of familiar types local martingale from the definition, quality, resolution, can anticipate to mean etc. begins and elaborate to talk about the necessity of local martingale. This chapter analysis the quality of convergence of local martingale specificly by approximation variation.The theory is applied to fulfillment, passes concrete present martingale of solid instance and the link of random partial differential equation in chapter 5, express martingale's importance in the finance markets realm. With the foudation of normal partial differential equation, we analysis the application of martingale in finance by adding a disturbance function, for example. economic crisis, government intervention etc.The last chapter is the foreground outlook of martingale theory and the prob-lems which need to settle.