Properties Of The Multidimensional G-expectation And Related Multidimensional Risk Measure

In this paper,we study some problems related to multidimensional expectation defined by multidimensional BSDE equation,and extend the results in the literatureSpecifically,the first chapter introduces the research background,current situation and significance of this paper,as well as some preparatory knowledge.In the second chapter,starting from the properties of one dimensional g-expectation we study some properties of multidimensional g-expectation,prove the constancy and monotonicity of expectation,and give the sufficient conditions of subadditivity,positive homogeneity and translation invariance of multidimensional expectation;In the rest of this chapter,we study the sufficient conditions of Jensen's inequality for multivariate concave convex functions based on multidimensional g-expectation,and extend the conclusion of Jensen's inequality to multidimensional case.Jensen inequality of multidimensional expectation is transformed into a case similar to Jensen inequality of one-dimensional expectation by matrix multiplication.In the third chapter,a quantitative multidimensional risk measure is based on the multidimensional g-expectation.It is proved that the quantitative multidimensional risk measure is well defined,that is,it satisfies zero value,monotonicity and translation invariance.On this basis,we study the multidimensional convex risk measure of quantity value and multidimensional consistent risk measure of quantitative value.In the fourth chapter,we get brief conclusions and prospects.