Study On The Properties Of G-expectation

Based on the notion of backward stochastic differential equation(denoted byBSDE),the properties of g-expectation, some properties of the g-expectation and theg-evaluation with anti-monotonic additivity(sub-additivity) were studied, Jensen’sinequality with the infinite time interval was also studied.In the first chapter, the BSDE’s background knowledge and the latest researchsta-tus were introduced.At the same time, the research methods and main results inthis thesis were presented.In the second chapter, the properties of g-expectation and some preliminarieswere introduced.In the third chapter, the properties of generator g and g-expectation were studied,with the condition that g-expectation meet anti-comonotonic additivity (subadditivity).The generator g is independent of y and is an odd function in z, if g-expectation isanti-comonotonic additivity. In addition that the conditional g-expectation also meetsanti-comonotonic additivity. The generator g is independent of y, if g-expectationmeets anti-comonotonic sub-additivity. Furthermore, if g is an odd function in z, thenthe conditional g-expectation meets anti-comonotonic sub-additivity, and thegenerator g is linear when the dimension d equals1.In the fourth chapter, some properties of g-evaluation and generator g wereresearched when g-evaluation meets anti-comonotonic additivity. If g-evaluation satis-fies anti-comonotone additivity, g is homogeneous in (y,z). Furthermore, the generatoris linear if the dimension dequals1.In the fifth chapter, Jensen’s inequality for g-expectation with infinite timeinterval was exploited.Aconclusion was obtained that Jensen’s inequality forg-expectation on the convex function holds in general if and only if g does not dependon y and g is a super-homogeneous generator in z.