Study On Martingale Problem Under Nonlinear Expectations And Related Problems

To deal with some uncertainty problems from the real world (e.g., Knightian un-certainty), Peng (2006)[52] introduced a new nonlinear expectation - G-expectation. In the recent decade, as a new branch of probability theory, nonlinear expectation theory, especially G-expectation theory, is developing very fast. G-expectation is a time consistent sublinear expectation constructed from a so called "G-heat equation": (?)tu-G((?)x2u)=0, meanwhile, G-Brownian motion is also introduced, subsequently the developing of stochastic calculus under G-expectation framework. We generalize the idea of constructing the G-expectation from a nonlinear partial differential equa-tion. By following Stroock and Varadhan’s (1969)[74] idea about establishing a direct connection between linear differential operators and probability measures when they studied diffusion processes, we connect nonlinear differential operators and nonlinear expectations, and introduce the notion of martingale problem under nonlinear expec-tations. To solve it, we follow the idea of Peng (2005)[51] for constructing nonlinear Markov chain to construct nonlinear expectations according to the viscosity solutions to nonlinear partial differential equations, and then prove the existence of the solu-tion. To this end, the existence and uniqueness of the viscosity solutions to a class of fully nonlinear partial differential equations are derived. Relying on the martingale problem approach, we introduce the notion of weak solution to G-stochastic differen-tial equation (G-SDE, for short), and prove the existence result for a class of general d-dimensional G-SDE with Lipschitz continuous coefficients. This result generalizes the classical Girsanov transformation in the sense that changing a nonlinear Brownian motion under one given nonlinear expectation to a nonlinear Brownian motion under the other nonlinear expectation.Firstly, the thesis reviews the classical martingale problem, and the motivation and development of the nonlinear expectation theory, and then the recent results and the de-velopment tendency. Secondly, we give the primary knowledge and some developed results on nonlinear expectation theory, especially the main results on stochastic calcu- lus under G-expectation theory. To construct general nonlinear expectations, we study the existence and uniqueness of the viscosity solutions for two large classes of fully nonlinear partial differential equations, and identify an appropriate class of fully non-linear partial differential equations for constructing nonlinear expectations. Thirdly, we construct a class of nonlinear expectations from the viscosity solutions of fully nonlin-ear partial differential equations, and study the properties of them. Then, we define the martingale problem under nonlinear expectations, and derive the existence of the problem with a special form. At last, we study the existence of the weak solution to general G-SDE by employing the martingale problem approach.