Diffusion Processes And Large Deviations Under Sublinear Expectations

Shige Peng introduced G-normal distributions, G-Brownian motions, G-expectations and the corresponding G-stochastic calculus through the non-linear heat equations, and developed the theory of sublinear expectation spaces. This new direction created by Peng provides a novel and initial stochastic analysis tool and view towards the solutions of non-linear problems, and it has tremendous applications in dynamic risk measures in finance fields.In this work, we mainly study the large deviations theory under sublinear expectations and the properties of diffusions solutions of stochastic differential equations driven by G-Brownian motion.This thesis is organized as follows:Chapter 1 is an introduction on the background, motivation, preliminaries and the main results of the dissertation.In chapter 2, we study large deviations and moderate deviations for independent ran-dom variables under sublinear expectations. By using subadditive method we establish the large deviation principles, and obtain the representation of the rate function through Varad-han asymptotical integral lemma under the capacity. The main tools to prove the moderate deviation principles are Peng's central limit theorem and truncation techniques.In chapter 3, we study large deviation principles for empirical measures of independent random variables under sublinear expectations. We first prove any finite dimensional large deviation principles and then establish the exponential tightness by constructing tight sets. Finally, by using representations theorem and minimax theorem we obtain a representations of the rate function.In chapter 4, we study the support of the diffusion solutions of the stochastic differential equations driven by G-Brownian motion. By defining suitable approximations and using the quasi invariant properties of G-Brownian motion and some moment estimates of G-stochastic integral, we obtain the support theorem of the solution of stochastic differential equations driven by G-Brownian motion.In chapter 5, we study the uniform large deviations and modulus of continuity of stochastic differential equations driven by G-Brownian motion. We first present a uniform exponential inequality of stochastic integral under G-expectations, then establish large deviation principle of G-stochastic differential equations locally uniform with respect to the initial values. Finally by using the above large deviation principle we obtain an upper bound of modulus of continuity of diffusion processes driven by G-Brownian motion.