Large And Moderate Deviations For Diffusion Process

The paper includes two parts. In the one part, our main purpose is to generalize the large deviation of 1-dimensional diffusion process to multidimensional case. For a multi-dimensional diffusion process dX(t) = σ(t)dB(t) with σ(t) unknown, we study the large and moderate deviations of the estimator Q_tⁿ(X) := of the quadretic variational process [X]_t = ∫₀^t (σσ^*)(s)ds. By Gartner-Ellis theorem, we can get the moderate deviation of the estimator above at the fixed time t = 1. By computing the Fenchel-Legendre transform of the logarithmic moment generating function, we get a explicit rate function. In the other part, For stationary Gaussian processes, we obtain the necessary and sufficient conditions for Poincare inequality and log-Sobolev inequality and provide the sharp constants. The extension to moving average processes is also presented.The paper is organized be four chapters: In Chapter 1, we summarize the background and development of the large deviation principle. In Chapter 2, we introduce simply the basic concepts and principles and usual methods of the LDP. In Chapter 3, we give the basic supposes and main lemmas of the LDP of diffusion process and we draw the main conclusions of the MDP of diffusion process at the fixed time t = 1. In Chapter 4, we give the main results about Poincare inequality and log-Sobolev inequality of stationary Gaussian process.