Multiscale Method For Stochastic Dynamical Systems:Averaging,Homogenization And Application

In many physical and financial situations,some of the variables are macroscopic or evolve slowly while others vary on smaller spacial or temporal scales,e.g.,hydrodynamical,molecular dynamical quantities in fluid mechanics and securities in financial market.One often tries to eliminate the fast variables in the slow system,and obtains a good approximation of the slow variables' evolution on a certain interval.This is the aim of multiscale method.This thesis is devoted to studying multiscale method for stochastic dynamical systems and is organized as follows.In Chapter 1,we introduce the background and researches of multiscale stochastic dynamical systems.In Chapter 2,we review some basic concepts and definitions about stochastic dynamical systems.In Chapter 3,mean exit time and escape probability for stochastic dynamical systems with oscillating coefficients are studied via homogenization.We analytically and numerically show that the homogenized systems faithfully capture these two dynamical quantities.Exacting data from observations on the mean exit time(or escape probability)of the original systems,we try to fit the mean exit time(or escape probability)of the homogenized systems by least square method.In this way,we can accurately estimate the unknown parameter under appropriate assumptions.In Chapter 4,we consider a multiscale dynamical system driven by ?-stable L(?)vy process.Under the some Lipschitz and bounded conditions,we establish the averaging principle.As an application,we apply SNM(stochastic Nelder-Mead)method to estimate the unknown parameter in the system.Finally,a summary and some questions that need to be impoved are presented.