Moment method estimation of first-order continuous-time bilinear processes

ABSTRACT

In the present paper, we propose an estimation method of the first order continuous-time bilinear (COBL) process based on Euler-Maruyama discretization of the Itô solution asociated with the stochastic differerential equation (SDE) defining the process, and we suggest a standard moment method (MM) estimates of the unknown parameters involving in COBL process. So, some relationships linking the parameters and the theoretical moments of the process and its quadratic version are given. These relationships we allow to construct two algorithms to estimate the parameters based on MM. Using the fact that the incremented processes are strongly mixing with exponential rate whenever certain conditions are fulfilled, we show that the resulting estimators are strongly consistent and asymptotically normal. The theory can be applied to the COGARCH(1, 1), Gaussian Ornstein-Uhlenbeck (OU) models and among other specifications. Finite sample properties are also considered throught Monte-Carlo experimencts. In end, this algorithm is then used to model the exchanges rate of the Algerian Dinar against the US-dollar and against the single European currency.

KEYWORDS:

• Asymptotic normality
• Consistency
• Continuous-time bilinear processes
• Quadratic processes

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 40A05
• 40A25
• Secondary 45G05

Acknowledgments

We would like to thank Prof. N. Balakrishnan editor of the journal, for his attention, encouragement and valuable advice. Also, we are very grateful to an anonymous referee for reading the paper very carefully and making many constructive remarks and suggestions.