Maximum Likelihood Estimation Of Stochastic Reaction-diffusion Equations

The research of stochastic (ordinary) differential equations started about one hundred year ago. In recent thirty years, with the development of differential equations, measure theory etc., more and more work on stochastic partial differential equations (SPDEs) appear and are applied in many fields, such as biology and physics, in which lots of important theory and results built Now SPDEs has been an important field.This thesis is devoted to the maximum likelihood estimates for a class of SPDEs, stochastic reaction and diffusion equation. Consider the following SPDEs defined on (θ,π) where 9 is unknown parameter and W is a Q-Wiener process, θ. We apply a finite dimensional approximation to give a finite dimensional maximum likelihood estimate θN for θ and then by using Ito formula and Burkholder-Davis-Gundy inequality and Markov inequality to show θN is consistency to 9.This thesis consisting of two parts:First we give an introduction some backgroud of SPDEs and some preliminary for our problem, this includes some basic knowledge on Hilbert space and infinite dimensional Wiener process and maximum likelihood estimates for stochastic differential equations. Then we study the stochastic reaction-diffusion equations and give the maximum likelihood estimates for θ and show the consistency.