Gaussian estimation for discretely observed Cox–Ingersoll–Ross model

This paper is concerned with the parameter estimation problem for Cox–Ingersoll–Ross model based on discrete observation. First, a new discretized process is built based on the Euler–Maruyama scheme. Then, the parameter estimators are obtained by employing the maximum likelihood method and the explicit expressions of the error of estimation are given. Subsequently, the consistency property of all parameter estimators are proved by applying the law of large numbers for martingales, Holder’s inequality, B–D–G inequality and Cauchy–Schwarz inequality. Finally, a numerical simulation example for estimators and the absolute error between estimators and true values is presented to demonstrate the effectiveness of the estimation approach used in this paper.

[1]  T. Deck.  Asymptotic properties of Bayes estimators for Gaussian Itô-processes with noisy observations , 2006 .

[2]  John J. McConnell,et al.  Valuation of GNMA Mortgage-Backed Securities , 1981 .

[3]  Gerhard-Wilhelm Weber,et al.  Parameter Estimation in Stochastic Differential Equations , 2012 .

[4]  D. Duffie,et al.  Modeling term structures of defaultable bonds , 1999 .

[5]  Jun Hu,et al.  Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements , 2013, Int. J. Control.

[6]  Mathieu Kessler.  Estimation of an Ergodic Diffusion from Discrete Observations , 1997 .

[7]  Maximum Likelihood Estimates of a Class of One‐Dimensional Stochastic Differential Equation Models From Discrete Data , 2001 .

[8]  M. N. Mishra,et al.  Approximate maximum likelihood estimation for diffusion processes from discrete observations , 1995 .

[9]  Jun Hu,et al.  Probability-guaranteed H∞ finite-horizon filtering for a class of nonlinear time-varying systems with sensor saturations , 2012, Syst. Control. Lett..

[10]  N. Yoshida.  Estimation for diffusion processes from discrete observation , 1992 .

[11]  Chenxu Li.  Maximum-likelihood estimation for diffusion processes via closed-form density expansions , 2013, 1308.2764.

[12]  Huijun Gao,et al.  Recent Advances on Recursive Filtering and Sliding Mode Design for Networked Nonlinear Stochastic Systems: A Survey , 2013 .

[13]  Stephen A. Ross,et al.  An Analysis of Variable Rate Loan Contracts , 1980 .

[14]  T. Alderweireld,et al.  A Theory for the Term Structure of Interest Rates , 2004, cond-mat/0405293.

[15]  M. Sørensen,et al.  Martingale estimation functions for discretely observed diffusion processes , 1995 .

[16]  Jun Hu,et al.  State estimation for a class of discrete nonlinear systems with randomly occurring uncertainties and distributed sensor delays , 2014, Int. J. Gen. Syst..

[17]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[18]  A. Basu.  13. Stochastic differential equations : The Bernstein-Von Mises theorem for a certain class of diffusion processes , 1984 .

[19]  Zidong Wang,et al.  Distributed state estimation for uncertain Markov‐type sensor networks with mode‐dependent distributed delays , 2012 .

[20]  Guoliang Wei,et al.  Reliable H∞ state estimation for 2-D discrete systems with infinite distributed delays and incomplete observations , 2015, Int. J. Gen. Syst..

[21]  Eduardo S. Schwartz,et al.  Analyzing Convertible Bonds , 1980, Journal of Financial and Quantitative Analysis.

[22]  Xiao-Xia Feng,et al.  Bayesian Estimation of CIR Model , 2012, Journal of Data Science.

[23]  F. H. M. Sanchez,et al.  Gaussian Estimation of One-Factor Mean Reversion Processes , 2013 .

[24]  Libo Sun,et al.  A penalized simulated maximum likelihood approach in parameter estimation for stochastic differential equations , 2015, Comput. Stat. Data Anal..

[25]  B. Prakasa Rao,et al.  ASYMPTOTIC STUDY OF THE MAXIMUM LIKELIHOOD ESTIMATOR FOR NON-HOMOGENEOUS DIFFUSION PROCESSES , 1985 .

[26]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .