| Fractional Brownian motion(FBM) attracts many scholars at home and abroad to conduct an in-depth study and then get a lot of results because of its unique nature. About 21 st century, the mathematical model of stochastic differential equations driven by fractional Brownian motion appeared in biology, finance and other fields.Because FBM is neither a Markov process, nor a semimartingale so that many random analysis stems from classic research techniques are not applicable, therefore it brings many difficulties in the research of fractional Brownian motion and their application. But it also allows fractional Brownian motion can be used to describe the process which Markov processes and semimartingale cannot describe. This made fractional Brownian motion has a wide range of applications in many fields. This paper is divided into five chapters, mainly discussing the generalized mixed fractional Brownian motion(GMFBM) linearly combined by multiple fractional Brownian motions and their MLE, as well as hypothesis testing for the estimators.In Chapter II we describe some preliminary knowledge, briefly presenting the Gerschgorin Circle theorem and its proof, Kolmogorov Regularity Theorem, Borel-Cantelli’s lemma, as well as the MLE with Brownian motion models.In Chapter III we present the GMFBM combined by multiple FBM, and explore its basic properties such as mixed-self-similarity, Correlation between the increments and its H?lder-continuity and α-differentiability of its sample paths, extending the corresponding conclusions of mixed fractional Brownian motion.In Chapter IV we study the MLE of parameter with mixed fractional Brownian motion model, study the statistical properties such as unbiasedness, consistency etc.In Chapter V, as for the maximum likelihood estimation given in the previous chapters, we use a likelihood ratio test to test the parameters in the model with generalized mixed fractional Brownian motion, and get a series of related results. |
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