the first order nonlinear autoregressive model ‎ ‎with ornstein uhlenbeck processes driven by white ‎noise

This paper presents a nonlinear autoregressive model with ‎ornstein ‎uhlenbeck processes innovation driven with white noise. ‎‎‎‎notations ‎and ‎preliminaries ‎are ‎presented ‎about ‎the ‎ornstein ‎uhlenbeck ‎processes ‎that ‎have ‎important ‎applications ‎in ‎finance. ‎the ‎parameter ‎estimation ‎for ‎these ‎processes ‎is ‎constructed ‎from ‎the ‎time ‎continuous ‎likelihood ‎function ‎that ‎leads ‎to ‎an ‎explicit ‎maximum ‎likelihood ‎estimator.‎ a semiparametric method is proposed to estimate the nonlinear autoregressive function using the conditional least square method for parametric estimation and the nonparametric kernel approach by using the nonparametric factor that is derived by a local l2-fitting criterion for the regression adjustment ‎estimation‎‎‎. then the ‎monte ‎carlo‎‎ numerical simulation studies are carried out to show the efficiency and accuracy of the present ‎work.‎ the ‎mean square error (‎mse) is a measure of the average squared deviation of the ‎estimated ‎function‎ values from the actual ones. the values of mse indicate ‎that ‎the ‎innovation ‎in ‎noise ‎structure ‎is ‎performed ‎well ‎in ‎comparison ‎with ‎the ‎existing ‎noise ‎in ‎the ‎nonlinear ‎autoregressive ‎models.‎‎‎‎