| Due to the actual needs of statistics in the process of finance,the basic theorems and frameworks of classical probability theory cannot solve the related problems of nonlinear risk measurement in finance and asset pricing in incomplete markets.Professor of Peng Shige for Shandong University put forward the basic concept and framework of sub-linear expectation space.Sub-linear expectation is a natural extension of classical linear expectation,which is widely used in statistics、finance、economy and measurement of risk.The proof of the limit theorem in classical probability space depends on the probability and the expectations additivity,but the capacity and expectation under sub-linear expectation spaces are no longer additive,many proof methods in classical probability spaces are no longer valid.This paper mainly studies the complete convergence、complete Choquet integral convergence、almost sure convergence and average convergence under sub-linear expectation space.Firstly,we prove complete convergence under sub-linear expectation spaces,compare to the complete convergence of the classical probability space,sub-linear expectations are not unique under the sub-linear expectation space,there are upper and lower expectations and the sub-linear expectation is defined on the local Lipschitz function,which adds difficulty to the proof of the sub-linear expectation space theorems.Secondly,when the convergence of the complete Choquet integral is proved,the integral is generally divided into two stages,where the proof of a piece of integral is generally based on complete convergence.Finally,it is prove that the average convergence and almost sure convergence of the sub-linear expectation space.The almost sure convergence under the sub-linear expectation space is defined by the convergence of the capacity.Capacity is divided into upper capacity and lower capacity,the almost sure convergence of the upper capacity can be pushed almost sure converges of the lower capacity,otherwise it does not hold.We prove that almost sure convergence under sub-linear expectation space is proof of the upper capacity of almost sure convergence. |
