Yule-Walker type estimator of first-order time-varying periodic bilinear differential model for stochastic processes

Abstract

This paper, studies the class of diffusion processes generated by a first-order continuous-time bilinear processes (COBL(1, 1)) with time-varying coefficients. So, we used the It$\widehat{o}$ formula approach for examining the ${\mathbb{L}}_2-$structure of the process and its powers. In time-invariant case, an expression of the moments of any order are given and the continuous autoregressive $\left(CAR\right)$ representation of such version is given, in particular the moments properties of some specifications are however derived. Based on these results we are able to examine the statistical properties as well as we develop an estimation method of the process via the so-called Yule-Walker (YW) type algorithm which relates with unknown parameters of CAR representation. The method is illustrated by a Monte Carlo study and applied to modeling the electricity consumption sampled at each 15 mn in Algeria.

Keywords:

• Diffusion processes
• Brownian motion
• Itô’s formula
• Yule-Walker estimates
• maximum likelihood estimates

Subject Classification:

• Primary 40A05
• 40A25
• Secondary 45G05

Acknowledgments

The authors would like to express their most sincere thanks and grateful acknowledgments to Professor Narayanaswa-my Balakrishnan, Editor-in-Chief for his considerable encouragement and to an anonymous Referee for his valuable remarks and pertinent suggestions, which were remarkably helpful in improving the content of the manuscript.