The Extension And The Research On The Analytic Properties About Ait-sahalia Rate Model

This thesis gives a framewoke on the interest rate, including the basic theory and the form of interest rate,the importance of research.At the same time,many stochastic differential equations have been used to describe the interest rate.Of course, the linear models hold "good" properties,the existence and uniquenese of the solution for the most equations. But nonlinear models don’t satisfy the linear growth condition or (locally) Lipschitz continuous,we should spend more time researching the existence and uniquenese of the solution and other analytic propertiesBased on the previous interest rate models, Ait-Sahalia model in particular,this thesis will propose the following highly nonlinear stochastic differential equations model on interest rate:dx(t)=(a_x-a (t)-ao+ax(t)-a2xa3(t))dt+(β0-β1xβ2(t)+β3xβ4(t))dw(t) where,a_>0,a>0,ai>0,i=0,1,2,a3>1, β0>0,β1>0,0<β2<1,β3>0,β4>1.We mainly do the following four work:Firstly, we prove that the equation has a unique global nonnegative solution.Secondly, we prove the boundedness of the solution including the first and the two order moment boundedness, the stochastic boundedness(Under certain conditions,the solution will converge to a bounded interval in probability,and the convergence is not affected by the initial value and the time T) and orbit estimation(solution of the rise and fall velocity properties).Thirdly, we prove that the Euler-Maruyama approximates converge to the true solution in probability.Fourthly, we use the obtained conclusion to compute the price of options and stock, at the same time,we give two examples.