Weighted G-Expectation And Related Properties

We research weighted g-expectation based on g-expectation. let λ∈[0,1],we denot εgλ[ζ]=λεg[ζ]+(1-λ)(-εg[-ζ]),(?)ζ∈L2(Ω,F,P), Which is called weighted g-expectations of ζ."Hurwitz optimism-pessimism approach" is our direct motivation of researching weighted g-expectation. A special circumstances which is meaningful is the generator g=μ|z|,μ is constant. For the g-expectation generated from this generator g,we know that there is a probability measure set S that depends only on the generator of g and we haveAccordingly, weighted g-expectations can be noted as At this point,weighted g-expected εgλ[ζ] can reflect expectation after the economic man giving appropriate weight to the most optimistic results and the most pessimistic results.From this instance, we can see that the weighted g-expectations εgλ[ζ] can better reflect different people’s expectations under the uncertain economic conditions.In Chapter l,we introduce research background and development of g-expectation and weighted g-expectations and research contents.In Chapter2, we introduce the basic theory of g-expectation and give the definition and basic properties of g-expectation and conditional g-expectation.We show the natures of g-expectation associated with the generator g.In Chapter3, variance, covariance, correlation coefficient is the fundamental concept of probability theory.We give correlation of random variable for weighted g-expectations,includi-ng variance, co variance, correlation coefficient for weighted g-expectations, And launch their fundamental nature. We draw the sufficient condition of which two random variables is positive (negative) correlation and an example of application for correlation coefficient.In Chapter4, we give some inequalities,law of large numbers and total monotonic, for weighted g-expectations. We firstly have weighted g-expectation increase with λ and get its properties.Then we have Chebyshev inequality for weighted g-expectation,which is used to pr-ove the law of large numbers.By Borel-Cantelli Lemma, lemma4.4.2and lemma4.4.3and we have important theorems:law of large numbers for weighted g-expectations.We have also studied basic properties of weighted g-expectation with comonotonic subadditivity.