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

TL;DR

This paper introduces a continuous-time GARCH(p,q) process with delay, derived as a limit of discrete GARCH models, and explores its mathematical properties, including existence, regularity, and stationarity.

Contribution

It generalizes the COGARCH process to include higher-order delays and provides a rigorous analysis of its existence, properties, and representations.

Findings

The process exists uniquely under certain conditions.

Path properties include positivity and piecewise differentiability.

The process exhibits mean and covariance stationarity under specific conditions.

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.