By choosing appropriate σ fields, incorporating the moment and measure of non orthogonality conditions, and using reverse martingale approach, the convergence on bivariate weighted s.

Using reverse martingale approach,we extend the results of arrays of row-column exchangeable random variables to exchangeable random variables,and discuss the convergence on bivariate weighted sums of triangular arrays of row-wise exchangeable random variables,and obtain some conclusions of the convergence on bivariate weighted sum of triangular arrays of row-wise exchangeable random variables.

The reversed martingale approach was used to obtain the law of the iterated logarithm for row-column exchangeable infinite arrays with finite (2 + δ) moments (δ>0).

Firstly,some properties of random essential supremum are discussed,set-valued inverse superpramart approximation and set-valued inverse supermartingale convergence theorem in the sense of Kuratowski are provided,respectively.

Some results are given for set-valued inverse supermartingale in the sense of Kuratowski-Mosco and weak limit.

The explicit expressions of some special equivalent martingale measure are given,such as the minimal entropy martingale measure,the minimal reverse relative entropy martingale measure,the variance-optimal martingale measure and the minimal martingale measure.

The method of martingale in Asian option pricing;

Φ-atomic Decompositions and Concave Φ-inequalities for Martingales;

The Chinese Stock Market Weak-form Efficiency Analysis——Martingale Method;

Relations of Stationary Processes, Martingales and Mardov Processes;

Fundamental Inequalities of Martingales on Weak L_p Spaces

Using martingales and random walks,the author got an approximate formula about probability of that the insurer′s surplus reaches an upper barrier before the ruin and its precise estimation.