The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to dZt=σ(Zt−)dXt
driven by a two-sided α-stable L´evy process, in the spirit of the classical Engelbert-Schmidt time-change  approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence and uniqueness of weak solutions for α∈(0,1). Our approach is not based on classical stochasticcal culus arguments but on the general theory of Markov processes. We prove integral tests for finiteness of path integrals under minimal assumptions. Keywords: Stochastic Differential Equations, stable processes, Markov processes, perpetuities, time change.
Original languageEnglish
JournalJournal of the European Mathematical Society
Publication statusPublished - 1 Jun 2023


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