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

Zanzotto, Pio Andrea

Ann. Probab., Tome 30 (2002) no. 1, p. 802-825 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2002-04-14
Classification:  strictly $\alpha$-stable Lévy processes,  Cauchy process,  stochastic differential equations,  weak existence,  "local" existence,  time change,  quadratic pure-jump semimartingales,  representation,  stable integrals,  60H10,  60J30