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April 2002 On stochastic differential equations driven by a Cauchy process and other stable Lévy motions
Pio Andrea Zanzotto
Ann. Probab. 30(2): 802-825 (April 2002). DOI: 10.1214/aop/1023481008

Abstract

We consider the class of one-dimensional stochastic differential equations

$$dX_t = b(X_{t-})dZ_t, \quad t \geq 0,$$

where $b$ is a Borel measurable real function and $Z$ is a strictly $\alpha$-stable Lévy process $(0 < \alpha \leq 2)$. Weak solutions are investigated improving previous results of the author in various ways.

In particular, for the equation driven by a strictly 1-stable Lévy process, a sufficient existence condition is proven.

Also we extend the weak existence and uniqueness exact criteria due to Engelbert and Schmidt for the Brownian case (i.e., $\alpha = 2$) to the class of equations with $\alpha$ such that $1 < \alpha \leq 2$. The results employ some representation properties with respect to strictly stable Lévy processes.

Citation

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Pio Andrea Zanzotto. "On stochastic differential equations driven by a Cauchy process and other stable Lévy motions." Ann. Probab. 30 (2) 802 - 825, April 2002. https://doi.org/10.1214/aop/1023481008

Information

Published: April 2002
First available in Project Euclid: 7 June 2002

zbMATH: 1017.60058
MathSciNet: MR1905857
Digital Object Identifier: 10.1214/aop/1023481008

Subjects:
Primary: 60H10 , 60J30

Keywords: "local" existence , Cauchy process , quadratic pure-jump semimartingales , representation , stable integrals , Stochastic differential equations , strictly $\alpha$-stable Lévy processes , Time change , weak existence

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 2 • April 2002
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