A Continuous Time GARCH(p,q) Process with Delay

Abstract: We investigate the properties of a continuous time GARCH process as the solution to a L\'evy driven stochastic functional integral equation. This process occurs as a weak limit of a sequence of discrete time GARCH processes as the time between observations converges to zero and the number of lags grows to infinity. The resulting limit generalizes the COGARCH process and can be interpreted as a COGARCH process with higher orders of lags. We give conditions for the existence, uniqueness and regularity of the solution to the integral equation, and derive a more conventional representation of the process in terms of a stochastic delayed differential equation. Path properties of the volatility process, including piecewise differentiability and positivity, are studied, as well as second order properties of the process, such as uniform L1 and L2 bounds, mean stationarity and asymptotic covariance stationarity.