Continuous-time GARCH process driven by semi-L\'evy process

TL;DR

This paper introduces a semi-Lévy driven continuous-time GARCH process, characterizing its properties, demonstrating its potential to approximate other processes, and analyzing its periodic stationarity and correlation structure.

Contribution

It develops the SS-COGARCH model driven by semi-Lévy processes, providing theoretical properties and demonstrating its ability to approximate a wide class of semi-Lévy driven COGARCH processes.

Findings

The state process converges almost surely.

The process exhibits periodic stationarity under certain conditions.

Increments are periodically correlated, confirmed by simulation tests.

Abstract

In this paper we study the simple semi-L\'evy driven continuous-time generalized autoregressive conditionally heteroscedastic (SS-COGARCH) process. The statistical properties of this process are characterized. This process has the potential to approximate any semi-L\'evy driven COGARCH processes. We show that the state representation of such SS-COGARCH process can be described by a random recurrence equation with periodic random coefficients. The almost sure absolute convergence of the state process is proved. The periodically stationary solution of the state process is shown which cause the volatility to be periodically stationary under some suitable conditions. Also it is shown that the increments with constant length of such SS-COGARCH process is itself a periodically correlated (PC) process. Finally, we apply some test to investigate the PC behavior of the increments (with constant length) of the simulated samples of proposed SS-COGARCH process.