Convergence Analysis For Numerical Solutions For Several Kinds Of Stochastic Economic Models

Stochastic differential equations have many applications in mathematics,physics and other fields.In recent decades,The research results of stochastic interest rate models are also fruitful.Both in the classical option pricing model and the CKLS model,the convergence of the numerical convergence problem of the value solution has gradually become one of the hot research problems.In particular,introducing the delay term makes the stochastic interest rate model more relevant to the fundamental interest rate changes,and thus the problem becomes more complex.The stochastic interest rate model describes the movements of stocks and securities in finance.By studying the nature of its solution,people can grasp the otherwise elusive trends of stocks and securities.The study of the convergence of the answer is,therefore,essential.This paper mainly gives the convergence of the numerical solutions of the Black-Scholers-Merton option pricing model and the CKLS time-delay stochastic interest rate model.The truncated Euler-Maruyama method needs to be constructed due to the effect of time delay.Firstly,we introduce the stochastic interest rate model’s background,development status,content and structure.The development and application of the Euler-Maruyama and truncated Euler-Maruyama methods in recent years are also presented.Secondly,the Black-Scholers-Merton equation is studied.The drift term and diffusion term in the equation satisfy Lipschitz condition and p-moment condition,and then the convergence analysis of the numerical solution of the model under the Euler-Maruyama method is given.Then the conclusions obtained are applied to specific cases to verify the validity of the findings.Finally,the constructed truncated Euler-Maruyama(EM)method is applied to the CKLS model with delayed volatility function,the existence of the solution of the model under local Lipschitz conditions and Khasminskii-type conditions is given,and then the convergence in probability of the numerical solution of the model under the same conditions is given.It is demonstrated that the approximate solution of the truncated EM method can be used in Monte Carlo for some numerically valued financial instruments,such as options and debt securities.