Annals of Applied Probability

Continuous-time GARCH processes

Peter Brockwell, Erdenebaatar Chadraa, and Alexander Lindner

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A family of continuous-time generalized autoregressive conditionally heteroscedastic processes, generalizing the COGARCH(1,1) process of Klüppelberg, Lindner and Maller [J. Appl. Probab. 41 (2004) 601–622], is introduced and studied. The resulting COGARCH(p,q) processes, qp≥1, exhibit many of the characteristic features of observed financial time series, while their corresponding volatility and squared increment processes display a broader range of autocorrelation structures than those of the COGARCH(1,1) process. We establish sufficient conditions for the existence of a strictly stationary nonnegative solution of the equations for the volatility process and, under conditions which ensure the finiteness of the required moments, determine the autocorrelation functions of both the volatility and the squared increment processes. The volatility process is found to have the autocorrelation function of a continuous-time autoregressive moving average process.

Article information

Ann. Appl. Probab., Volume 16, Number 2 (2006), 790-826.

First available in Project Euclid: 29 June 2006

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Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 60G12: General second-order processes 91B70: Stochastic models
Secondary: 60J30 60H30: Applications of stochastic analysis (to PDE, etc.) 91B28 91B84: Economic time series analysis [See also 62M10]

Autocorrelation structure CARMA process COGARCH process stochastic volatility continuous-time GARCH process Lyapunov exponent random recurrence equation stationary solution positivity


Brockwell, Peter; Chadraa, Erdenebaatar; Lindner, Alexander. Continuous-time GARCH processes. Ann. Appl. Probab. 16 (2006), no. 2, 790--826. doi:10.1214/105051606000000150.

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