Stochastic Differential Equation With Reflecting Boundary Condition

The thesis is a review,we summarize the background,research methods and results of Skorokhod problem and Skorokhod equation between 1950s to 2000,both of which are stochastic differential equations with reflected boundary condition.The main goal is summarizing the results and methods of four kinds of problems which are Skorokhod problem with normal reflecting boundary condition,Skorokhod euqation with normal reflecting boundary condition,Skorokhod problem with obique reflecting boundary condition and Skorokhod equation with oblique reflecting boundary condition.The structure of the article is the following:In the first chapter,we introduce the background of stochastic differential equa-tion with reflecting boundary condition briefly and illustrate the subject of the article.We particularly introduce the background of Skorokhod problem and Skorokhod eu-qation in the second chapter.It originated by the research of Skorokhod(23;24),then Tanaka(26),Lions and Sznitman(17),Saisho(20),Dupuis and Ishii(6)and many other mathematicians studied stochastic equation in multidimensional cases with normal or oblique reflection direction.In the third chapter,we summarize four kinds of Skorokhod problems or Skorokhod euqation.We also give out the definition of the aera condition and reflection direction.The main result are given in this chapter as well.In the fourth chapter,we list out lemmas and propositions needed to prove the main results.Chapter five is to prove four main theorems stated in Chapter three,using the classic fixed point method,penalty method,induction method and uniform approximation.Aftermost,we give out the research summary of our work.