A Nonlinear Interest Model With Time Delays And Markovian Switching And The Convergence Of Its Numerical Solutions

In 1996,Ait Sahalia introduced a highly nonlinear interest-rate model that can capture the complicated oscillation of the interest rate in some financial markets.How-ever,many of the data and research in financial market show that the interest rate is not only affected by the current information but also influenced by the historical infor-mation.In addition,as the economy and social environment changes,the parameters in the model should be changed as well.In this thesis,we extend the classic Ait-Sahalia model to the case with time delay and Markovian switchingSince the model is used to describe the oscillation of interest rate,it is required in practice that the solution should be positive.We can see that neither the drift part or the diffusion part of the model satisfies the global Lipschitz condition or linear growth condition,in addition,the drift part is highly nonlinear,hence the classic way of proving the existence,uniqueness and positivity of the solution cannot be employed here directly.In this thesis,we consider the parameters satisfy the conditions of ?1>1,72>0,2?1+ 6?2<3.By using different techniques say,generalized Ito formula,the boundedness of polynomial,Holder inequality and other elementary inequalities,we prove several vital properties of the solution,for instant the existence and uniqueness of the solution,positivity of the solution,the pth moment boundedness,stochastic boundedness and pathwise asymptotic estimate.Due to the complexity of the Ait-Sahalia model,it is not easy to obtain the ex-plicit solution.Hence the numerical solution is used here to approximate the underlying solution.The priority problem of numerical method is its convergence.Under the con-ditions of 71>1,72>0,271 + 6?2<3,we prove that the Euler-Maruyama solution converges to the underlying solution in probability.The techniques we used here in-clude the stopping time,Burkholder-Davis-Gundy inequality and several theorems in stochastic analysis.