Dynamics And Statistics Analysis Of Stochastic Diferential Equations Driven By Fractional Brownian Motion

Stochastic diferential equations are more accurate and realistic to describe real worldthan ordinary diferential equations, since real world is inevitably afected by some randomor uncertain factors. Recently, numerous fact demonstrated that the random factors havelong-range memory and dependent increments, and further showed that the fractionalBrownian motion or stochastic diferential equation driven by fractional Brownian motionis one of the most efective approaches describing such phenomena. Consequently, detailedresearch on stochastic diferential equations driven by the fractional Brownian motion isnot only necessary but also significance in theory and application.In this dissertation, the reducibility, stochastic stability, stochastic bifurcation andstatistical properties of stochastic diferential equations driven by the fractional Brownianmotion and dynamics of infinity dimensional stochastic evolution equation are studiedextensively. The detailed framework of this dissertation is as follows.Chapter1gives a survey to the background of this dissertation, the definition andsome basic properties of the fractional Brownian motion, stochastic diferential equationsdriven by the fractional Brownian motion, and the development of stochastic stabilityand stochastic bifurcation.Chapter2studies the necessary and sufcient conditions to reduce the generalstochastic diferential equations driven by some fractional Brownian motion. By usingthe theory on Wick product, we derive a general fractional It o formula, and apply it to ageneral stochastic diferential equations driven by the fractional Brownian motion, thenwe get some necessary and sufcient conditions of reducibility and the explicit solution.Moreover, the obtained results are applied to the one-factor model of a mean-revertingcommodity price and the stochastic population model driven by a fractional Brownianmotion, respectively. This improves and generalizes the corresponding results of the It ostochastic diferential equations.Chapter3considers stochastic stability of the general stochastic diferential equa-tions driven by fractional Brownian motion. Similar to the It o stochastic diferentialequations, we establish some suitable stochastic Lyapunov function and define a newclass of derivative operator, thus we obtain the judgement theorems for stochastic sta-bility, stochastic asymptotic stability, stochastic asymptotic stability in the large, andpth moment exponential stability of a general stochastic diferential equations driven byfractional Brownian motion. Moreover, the obtained results are applied to the Ornstein-Uhlenbeck process driven by a fractional Brownian motion to verify the criteria of the mentioned four types of stochastic stability, respectively.Chapter4considers stochastic stability and stochastic bifurcation of the Black-Scholes model driven by a fractional Brownian motion. Combining the expression ofthe explicit solution and the corresponding Lyapunov exponents, we obtain the necessaryand sufcient conditions for almost sure asymptotic stability and pth moment asymptoticstability, and further get the condition for generating the large deviations phenomenon.Moreover, we analyze the condition for generating the stochastic bifurcation of this model.Noting that it is the first time to study the stochastic bifurcation under the frameworkof the stochastic diferential equations driven by fractional Brownian motion.Chapter5studies the statistical properties of the Ornstein-Uhlenbeck process drivenby a fractional Brownian motion. Based on the corresponding Fokker-Planck equationfor the stochastic diferential equations driven by the fractional Brownian motion, wedeal with the probability density function of the Ornstein-Uhlenbeck process driven bythe fractional Brownian motion, and obtain that the mean square displacement decaysas power law. Therefore, it can describe the anomalous difusion.Chapter6considers the stochastic evolution equation driven by an infinite dimen-sional fractional Brownian motion. By using the stochastic analysis theory and the in-equality method, we rigorously prove the existence and uniqueness of variational solutionsto such system and then prove this variational solution can generate a continuous randomdynamical system.