Study On Some Problems Of Weighted G-Expectation And Backward Doubly Stochastic Differential Equations

This paper mainly studies on some problems of weighted g-expectation and Back-ward Doubly Stochastic Differential Equations.In chapter 1,we introduce some backgrounds and research status and main contents.The basic concepts and related properties of g-expectation are also briefly introduced,which provide theoretical knowledge for research.In chapter 2,we research translation invariance,quasi-subadditivity and monotonic-ity on A for weighted g-expectation,which is defined on the notation of g-expectation.Jensen's inequality,Moment inequality and law of large numbers for weighted g-expectation can be obtained.We prove that if the generator g is non-increasing with respect to g and positive-homogeneous with respect to(y,z),the Moment inequality for weighted g-expectation holds in general.When ? ? 1/2 and the generator g is indepen-dent of g,we establish Jensen's inequality for weighted g-expectation when g is super-homogeneous with respect to z,and we establish the law of large numbers for weighted g-expectation when g is sublinear with respect to z.In chapter 3,We investigate the properties of weighted g-variance and weighted g-covariance,and relationships between weighted g-variance and weighted g-covariance and generators g.Meanwhile,the relationships between weighted g-variance,weighted g-covariance and classic mathematical variance are showed.In chapter 4,we aim to investigate the uniqueness of minimal solution of Backward Doubly Stochastic Differential Equations,where the generator/is continuous and has a linear growth in(y,z),and the generator g satisfies Mao's non-Lipschitz condition in(y,z).