Evolutionary transfer functions solution for continuous–time bilinear stochastic processes with time-varying coefficients.

Abstract

In the present paper we study some probabilistic and statistical properties of continuous-time version of the well-known bilinear processes driven by standard Brownian motion. This class of processes, which includes many popular processes, was defined as a non linear stochastic differential equation which has increased rapidly in recent years, largely because of the high-frequency data available in many applications, particularly in finance and turbulence. We deal with the frequency domain, i.e., using the Wiener–Itô spectral representation to describe the second-order properties. So, we construct a recursive evolutionary transfer function system and study the ${\mathbb{L}}_2-$ structure of the process and its powers. The main stylized facts (unpredictability of model, and hence predictability of squared one, Taylor property, leptokurticity of the marginal distributions, Leverage effect, etc.) which are common to a large number of financial series and are difficult to reproduce artificially using stochastic differential equation are presented.

KEYWORD:

• Continuous-time bilinear processes
• Wiener-Itô representation
• Evolutionary transfer functions
• Taylor property

2010 AMS Mathematics Subject Classification:

• Primary 40A05
• 40A25
• Secondary 45G05

Acknowledgments

The authors would like to express their most sincere thanks and grateful acknowledgments to Professor Narayanas-wamy Balakris-hnan, 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.