AISM 53, 113-124

© 2001 ISM

## Lévy-driven CARMA processes

### P. J. Brockwell

Statistics Department, Colorado State University,
Fort Collins, Colorado 80523-1877, U.S.A.

(Received April 25, 2000; revised June 23, 2000)

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

Key words and phrases:
Lévy process, CARMA process, stochastic
differential equation, stable process.

**Source**
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