The Quadratic Covariation Of G-Brownian Motion And Related Processes

This dissertation aims to study quadratic covariation and related topics of the G-Brownian motion and related processes.Firstly,let L be the local time of G-Brownian motion B.We prove the existence of the integral (?) where f is of bounded p-variation with 1<p<2.Furthermore we establish the generalized Ito formula (?),where F? C1(R)and F' = f.Under some suitable conditions we show that the Bouleau-Yor identity(?) tholds,where<f(B),B>is the quadratic covariation of f(B)and B under sublinear expectation.Next,for G-Brownian motion B,we consider the functionalet(?),where v.p.stands for Cauchy's principal value.We show that the functional is well-defined for all a ? R and et(·)coincides with the Hilbert transform of the local time L(·,t)of G-Brownian motion B for every t.As a natural result we get a generalized Ito formula(the sublinear version of Yamada's formula)Btlog|B|l-Bt = ?0tlog|Bs|dBs + 1/2 et(0),where the integral is the Ito integral.The functional e(a)is a special case of the functionals of the form (?),where F is an absolutely continuous function such that the second derivative F" exists in the sense of Schwartz's distribution.Finally,we consider a bi-fractional Brownian motion BH,K with indices H ?(0,1),K ?(0,1],2HK = 1 and let L(x,t)be its local time process.We con-struct a Banach space H of measurable functions such that the quadratic covariation[f(BH,K),BH,K]and the integral fR f(x)L(dx,t)exist provided f?H.Furthermore,we prove the generalized Ito formula:(?)where the integral ?f0.f(BsH,K)dBsH,K is the Skorohod integral,f ?H is left continuous with right limit and F is an absolutely continuous function with d/dxF =f.Moreover,the Bouleau-Yor identity (?) holds for all f?H.