## The Annals of Probability

### On stochastic differential equations driven by a Cauchy process and other stable Lévy motions

Pio Andrea Zanzotto

#### 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.

#### Article information

Source
Ann. Probab. Volume 30, Number 2 (2002), 802-825.

Dates
First available in Project Euclid: 7 June 2002

http://projecteuclid.org/euclid.aop/1023481008

Digital Object Identifier
doi:10.1214/aop/1023481008

Mathematical Reviews number (MathSciNet)
MR1905857

Zentralblatt MATH identifier
1017.60058

#### Citation

Zanzotto, Pio Andrea. On stochastic differential equations driven by a Cauchy process and other stable Lévy motions. Ann. Probab. 30 (2002), no. 2, 802--825. doi:10.1214/aop/1023481008. http://projecteuclid.org/euclid.aop/1023481008.

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• VIA TOMADINI, 301A 33100 UDINE ITALY E-MAIL: zanzotto@dm.unipi.it