Abstract
Motivated by the need to describe regime switching in stock prices, we introduce and study a stochastic process in continuous time with two regimes and one threshold driving the change in regimes. When the difference between the regimes is simply given by different sets of real-valued parameters for the drift and diffusion coefficients, we show that there are consistent estimators for the threshold as long as we know how to classify a given observation of the process as belonging to one of the two regimes.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Brockwell, P.J.: On continuous time threshold ARMA processes. J. Statist. Plann. Inference 39, 291–303 (1994)
Brockwell, P.J., Stramer, O.: On the approximation of continuous time threshold ARMA processes. Ann. Inst. Statist. Math. 47, 1–20 (1995)
Brockwell, P.J., Williams, R.J.: On the existence and application of continuous-time threshold autoregressions of order two. Adv. Appl. Prob. 29, 205–227 (1997)
Chan, K.S.: Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Ann. Stat. 2(1), 520–533 (1993)
Chan, K.S., Tsay, R.S.: Limiting properties of the least squares estimator of a continuous threshold autoregressive model. Biometrica 85(2), 413–426 (1998)
Freidlin, M., Pfeiffer, R.: A threshold estimation problem for processes with hysteresis. Finance. Stochast. 36, 337–347 (1998)
Hansen A.T., Poulsen, R.: A simple regime switching term structure model. Finance. Stochast. 4(4), 409–429 (2000)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991)
Lamberton, D., Lapeyre, B.: Introduction to Stochastic Calculus Applied to Finance, 2nd edn. Chapman and Hall/CRC, Boca Raton (2008)
Mota, P.P.: Brownian motion with drift threshold model. PhD dissertation, FCT/UNL (2008)
Øksendal, B.: Stochastic Differential Equations. Springer, New York (2007)
Petrucelli, J.D.: On the consistency of least squares estimators for a threshold AR(1) model. J. Time. Anal. 7(4), 269–278 (1986)
Tong, H.: Non-linear Time Series: A Dynamical System Approach. Oxford University Press, Oxford (1990)
Acknowledgements
This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through PEst-OE/MAT/UI0297/2011 (CMA).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mota, P.P. (2013). On a Continuous-Time Stock Price Model with Two Mean Reverting Regimes. In: Lita da Silva, J., Caeiro, F., Natário, I., Braumann, C. (eds) Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications. Studies in Theoretical and Applied Statistics(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34904-1_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-34904-1_31
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34903-4
Online ISBN: 978-3-642-34904-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)