The Distribution Properties Of Stochastic Differential Equation Under G-Expectation

In this thesis,we investigate Wang's Harnack and shift Harnack inequalities for several classes of stochastic differential equations(SDEs,in short)under nonlinear expec-tation framework.Moreover,we characterize the path independence of additive function-als for SDEs driven by the G-Brownian motion(G-SDEs,in short)by using a class of nonlinear partial differential equation(PDE,in short).In addition,we discuss the order preservation for path dependent G-SDEs.The remainder of the paper is organized as follows.In the first part,we outline the background of the research and recall the prelimi-nary for dimension-free Harnack inequality,G-expectation,G-Brownian motion and G-Girsanov's transform.In the second part,we establish Harnack and shift Harnack inequalities for G-SDEs by means of coupling by change of measure,which the noise is additive.The results generalize the ones in the linear expectation setting.Moreover,some applications in quasi-invariant expectations are also given.In the third part,the Harnack and log-Harnack inequalities for G-SDEs with multi-plicative noise are derived by means of coupling by change of measure,which extend the correspongding results derived in[78,Theorem 3.4.1,Chap.3]under the linear expecta-tions.Moreover,the gradient estimate under nonlinear expectations is obtained.In the fourth part,the Harnack and shift Harnack inequality for G-SDEs with de-generate noise are derived by method of coupling by change of measure.Moreover,the gradient estimate for the associated nonlinear semigroup Pt|?Ptf|?c(p,t)(Pt|f|p)1/p,f?Cb+(Rd),p>1,t>0 is also obtained.As an application of Harnack inequality,we prove the weak existence of degenerate G-SDEs under some integrable conditions.Finally,an example is presented.In the fifth part,the path independence of additive functionals for SDEs driven by the G-Brownian motion is characterized by a class of nonlinear PDEs.Then,a sufficient and necessary condition for the path independence of As,t f,g is obtained,so that main results in[90,71,88,89]are extended to the present nonlinear expectation setting.In the end,sufficient and necessary conditions are presented for the order preservation of path dependent G-SDEs.Differently from the corresponding study of path independent G-SDEs,we use a probability method to prove the results.Moreover,the results extend the ones in the linear expectation case and in the G-SDEs without delay.