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Lévy-driven CARMA processes. (English) Zbl 0995.62089

Summary: Properties and examples of continuous-time ARMA (CARMA) processes driven by Lévy processes are examined. By allowing Lévy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Lévy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process.

We make use of the properties of general Lévy processes to investigate CARMA processes driven by Lévy processes $\left\{W\left(t\right)\right\}$ without the restriction to finite second moments. We assume only that $W\left(1\right)$ has finite $r$-th absolute moment for some strictly positive $r$. The processes so obtained include CARMA processes with marginal symmetric stable distributions.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 60G51 Processes with independent increments; Lévy processes 60H10 Stochastic ordinary differential equations
##### Keywords:
Levy processes; stable processes; CARMA processes