Studies On Several Types Of Spaces For Martingale Theory

Martingale space theory is a branch of mathematics that is closely related to functional analysis,stochastic analysis and harmonic analysis.After more than half a century of research,it has not only systematic theory,but also extensive applications in financial mathematics,risk analysis,stochastic control theory and other disciplines.In this doctoral dissertation,we mainly investigate about several types of spaces for martingale theory via stopping time,atomic decomposition and analysis techniques.The thesis consists of four chapters which are outlined in the following:In the first chapter,the research background is introduced on martingale and martingale space theory.Moreover,the history and current development of martingale space theory in frameworks of Lorentz space and Grand Lebesgue space are briefly reviewed.Finally,the main work and some preliminary tools used in the proof of this dissertation are outlined.In chapter 2,the theory of Orlicz-Lorentz Hardy martingale spaces are de-veloped,which are much more wider than the classical Lorentz Hardy martingale spaces.More precisely,several basic properties of Orlicz-Lorentz spaces are investi-gated,and then construct the atomic decomposition theorems of these martingale function spaces.Als07 the comprehensive description of the dual theorem of Orlicz-Lorentz Hardy spaces for martingales is introduced.Furthermore,the boundedness of generalized fractional integral operators I? in this new framework are studied,where ? is a non-negative concave function.In chapter 3,Lorentz martingale spaces with variable exponents on probability spaces are introduced.We give the John-Nirenberg inequality in this framework,which is the non-rearrangement invariant Banach function space,relying on the tool of atomic decomposition.More precisely,if the stochastic basis {F}n?0 is regular,then for any suitable 1?p(·)<? and 0<q<? we haveBMOp(·),q=BMO.In chapter 4,three types of Grand-martingale spaces are investigated.Based on the definitions of Grand Hardy martingale spaces,Doob's maximal inequality,atomic decomposition and the John-Nirenberg inequality on this type of martingale spaces are established.Moreover,we introduce Grand-Morrey martingale spaces and generalized Grand-Morrey martingale spaces.Doob's maximal inequalities,and the boundedness of fractional integrals as martingale transforms on these spaces are presented.