G-variance And Related Properties

As a basic tool for researching random phenomenon, the classic mathematicalexpectation theory has been improved quickly. With the development of finance andeconomy, a lot of uncertainty phenomenon without additive property can not be exactlydescribed by classic mathematical expectation theory as a result of linearity. The g-expectation based on the backward stochastic differential equations (BSDEs for short)maintains most of good properties except the linearity. Thus it can be used as a tool todescribe such uncertainty phenomenon precisely.In this paper we mainly investigate some properties of g-variance. By the methodof properties of g-expectation, classic expectation and comparison theorem of BS-DEs, we present some relations between g-variance and classical variance, g-varianceVarg(X):=εg[(X ε2g[X])] and D2g(X):=(εg[X])2εg[X], g-variance and g-covariance, and the equivalence relation between g-variance and generator g. Thispaper is organized by5chapters and the main contents are as follows:Chapter1introduces some backgrounds and research status about g-expectationand g-variance. The main contents and preliminaries are also briefly introduced.Chapter2gives a detailed description of definitions and properties about g-expectation and g-covariance, which provides theoretical knowledge for research.Chapter3is devoted to show the relationship between g-variance and classic vari-ance via properties of classic expectation and g-expectation as well as comparison the-orem of BSDEs.Chapter4presents relationship between g-variance Varg(·) and operator Dg(·)when generator g satisfies some certain conditions. Meanwhile, the equivalence rela-tion between g-variance and g-covariance, g-variance and generators are showed.Chapter5summarizes some techniques used in this thesis and innovations, andgives some ideas for future research.