| Probability theory focuses on studying the regularity of random phenomenon, which is extensively used in natural science, social science, practical production and so on. The large deviation theory beginning at 1960’s, studies the problem of the ergodic convergence speed. Nowadays, it has been one of the major branches in probability theory and has many applications in partial differential equations, Markov processes, statistics, insurance and finance.This article includes the following four chapters.In the first chapter, at first, we briefly review the basic concepts and results in large deviation theory. Then some known conclusions are stated. Finally, we give the motivations of this thesis.In the second part, the model studied in the paper is introduced, and the results obtained in this thesis are stated. As an application, asymptotic properties for parameter estimators and hypothesis testing in several types of parabolic stochastic partial differential equations are investigated.In the third chapter, we give the proof of the moderate deviation for the parameter estimator qùwhen the dimension N is fixed and observation time T tends to infinity. As it’s known, the difference between qùand its true value is a normalized martingale. Therefore, at first, we use the transformation of measures to verify the exponential equivalence between the quadratic variation process and its expectation. Then the moderate deviations of the martingales and splitting techniques are applied to conclude the moderate deviations for qùIn the fourth chapter, we consider the following two situation:N is fixed, T ? ￥ and T is fixed, N ? ￥.Rejection regions in the hypothesis testing are constructed respectively according toNeyman-Pearson theory. Then using the moderate deviation of the logarithmic likelihood ratio, theparameters in the rejection regions are calculated. Moreover, two types of errors decay to zero atexponential rates. |
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