On stochastic differential equations driven by a Cauchy process and other stable Lévy motions
The author considers existence and uniqueness of weak solutions of the one-dimensional stochastic differential equation \[ X_t = X_0 + \int_{]0,t]} b(X_{s-}) dZ_s, \quad t \geq 0, \] where \(X_0\) has an arbitrary distribution, \(b\) is a Borel measurable real function, \(Z\) is a strictly \(\alpha\)-stable Lévy process, \(0 < \alpha \leq 2\), \(X_0\) and \(Z\) are stochastically independent, and (when \(\alpha < 2\)) either \(Z\) is symmetric or \(b\) is nonnegative. The weak existence and uniqueness exact criterion of Engelbert and Schmidt for \(Z\) Brownian motion, i.e.\ \(\alpha=2\), is extended to \(\alpha\)-stable processes with \(1 < \alpha \leq 2\). This criterion is given in terms of the zero set and the \(\alpha\)-singularity set for the coefficient \(b\). Sufficient conditions based only on the coefficient \(b\) are given for the existence of a solution in the cases that \(Z\) is a Cauchy process, that \(b\) is nonnegative and \(Z\) is an \(1\)-stable Lévy process, and that \(Z\) is an \(\alpha\)-stable Lévy process with \(0 < \alpha < 1\). A local version (LH)(x) of the condition (H)(x) in the paper of the author [Stochastic Processes Appl. 68, No.~2, 209-228 (1997; Zbl 0911.60037)] is introduced, and the meaning of the condition (H)(x) is clarified. Sufficient conditions for the existence of nontrivial local weak solutions starting from a fixed point \(X_0 \equiv x\) are given in terms of the condition (LH)(x). The proofs rely on a representation property with respect to strictly stable Lévy processes.
scientific article; zbMATH DE number 1906069
scientific article