Continuous Time Autoregressive Moving Average Processes Driven by Semi-Levy Process

A flexible and tractable class of linear models is Autoregressive moving average (ARMA) process that are in effect of discrete noises.  The continuous time ARMA (CARMA) processes have wide applications in many data modeling where are more appropriate than discrete time models [1]. Specifically when the processes include high frequency, irregularly spaced data and or have missing observations. Many of these data show periodic structure in their squared log intraday returns [2]. In financial markets, variations and jumps play a critical role in asset pricing and volatilities models. The Levy-driven versions of these processes studied in [3]. The back-driving Levy process has two main components, the continuous variations part and the pure jump component [4]. The Levy-driven CARMA process described as the unique solution of some stochastic differential equation [5].  It is known that these family of CARMA processes are stationary or asymptotic stationary. The Levy processes have stationary increments while semi-Levy process have periodically stationary increments and are more realistic in many cases.  In this article, we study the semi-Levy driven CARMA processes.  We study the case where the back driving process is semi-Levy compound Poisson process.

Semi-Levy CARMA Process:

Presenting the structure of the semi-Levy processes and their characterization, we show that the semi-Levy driven CARMA process has periodic mean and covariance function.  To show this, we present some proper discretization for the process in which successive period intervals where the   period interval is  where  is the period. Then consider some predefined partition of all period intervals consist of   subintervals with different length but are the same for all period intervals.  The jump processes, say Poisson process, assumed to has fixed intensity parameter on each subinterval, say on  subinterval of each period interval, so has periodic property .  Then the semi-Levy compound Poisson process is defined by  where  is the semi-Levy Poisson process,  is some positive constant and the jumps with size  are iid random variables.  The state representation of the process is where the state equation is .We present the theoretical results and prove the periodically correlated structure of the process.We also investigate periodically correlated behavior for the simulated data  of the model.  Simulating the underlying measure and using discretization with 12 equally space samples in each period interval of the process, we divide the samples into corresponding 12 dimensional process for checking their stationarities.  Then we present the plot of the correlogram and the box plot of the corresponding multi-dimensional stationary processes and also corresponding cross-correlograms. The stationarity of these correspondence multivariate processes illustrates how this class of CARMA process is periodically correlated. 

The following conclusions were drawn from this research.The theoretical structure and state space representation of  CARMA process driven by semi-Levy compound Poisson process are obtained.The statistical properties and characteristics of the process are presented and it is shown that the process have periodically correlated structure.By simulated data and plotting the correlograms  and Box-plots for corresponding multi-dimensional process for the equally space discretization sample,  the periodic behavior of the process is verified.