Cramér Type Moderate Deviations Of Log-likelihood Ratio For Diffusion Processes

This paper mainly focuses on Cramer type moderate deviations based on the log-likelihood ratio of general diffusion processes.The diffusion process includes Ornstein-Uhlenbeck type processes and α-Brownian bridge.This paper is divided into six chapters:The first chapter is an introduction.Firstly,it introduces the research process and background of the topic.Secondly,the current research status of moderate deviation theory,especially Cramer type moderate deviation is analyzed.Finally,the research ideas and main work of this paper are introduced.The second chapter introduces the basic knowledge required for this paper.The third chapter studies the Cramer type moderate deviation of α-Brownian bridge.Ifα>1/2,the process is stationary.This chapter mainly uses mod-φ convergence method to obtain the Cramer type moderate deviations of α-Brownian bridge log-likelihood ratio through the logarithmic moment generating function,and the decay rate of the error probability of the hypothesis testing problem is given.The fourth chapter studies the Cramer type moderate deviation of the log-likelihood ratio process in a stationary state for the general diffusion process dXt=ab(t)Xtdt+σ(t)dWt,X0=0,and we discuss the cases of sign(α-K)=sign(K),K≠0 and K=0,b(t)=σ2(t)/2κ,κ>0,and the decay rate of the error probability for the hypothesis testing problem is given.The fifth chapter studies the Cramer type moderate deviation of the log-likelihood ratio process in non-stationary states for general diffusion processes and also discusses the cases of sign(α-K)=-sign(K),K≠0 and K=0,b(t)=σ2(t)/2κ,κ>0.This chapter uses the deviation properties of multiple Wiener Ito integral,exponential equivalent estimation and the asymptotic analysis techniques to obtain Cramer type moderate deviation results,and we apply the obtained results to hypothesis testing problems.The sixth chapter summarizes the paper and prospects for future work.