# Semi-Levy driven continuous-time GARCH process

## TL;DR

This paper introduces the semi-Levy driven continuous-time GARCH (SLD-COGARCH) process, characterizing its properties, stationarity conditions, and periodic correlation, with applications to high-frequency financial data modeling.

## Contribution

The paper develops the SLD-COGARCH model, providing theoretical conditions for stationarity and demonstrating its periodic correlation properties and practical data fitting.

## Key findings

- SLD-COGARCH process exhibits strict periodic stationarity under certain conditions.
- Increments of SLD-COGARCH are periodically correlated, confirmed by simulation tests.
- Model effectively fits high-frequency financial data.

## Abstract

We study the class of semi-Levy driven continuous-time GARCH, denoted by SLD-COGARCH, process. The statistical properties of this process are characterized. We show that the state process of such process can be described by a random recurrence equation with the periodic random coeffcients. We establish sufficient conditions for the existence of a strictly periodically stationary solution of the state process which causes the volatility process to be strictly periodically stationary. Furthermore, it is shown that the increments with constant length of such SLD-COGARCH process are themselves a discrete-time periodically correlated (PC) process. We apply some tests to verify the PC behavior of these increments by the simulation studies. Finally, we show that how well this model fits a set of high-frequency financial data.