On One-Dimensional Stochastic Equations Driven by Symmetric Stable Processes

Abstract We study stochastic equations Xt = xo + fQ b(u,Xu-) dZu, where Z is an onedimensional symmetric stable process of index a with 0 < a < 2, b : [0, oo) x R —> R is a measurable diffusion coefficient, and XQ e R is the initial value. We give sufficient conditions for the existence of weak solutions. Our main results generalize results of P. A. Zanzotto [20] who dealt with homogeneous diffusion coefficients b. In the nonhomogeneous case we present new sufficient conditons for the existence of (nonexploding) solutions even if Z is a Brownian motion. Using the property that appropriate time changes of stochastic integrals with respect to stable processes are again stable processes with the same index, we present a new proof of the main result which simplifies the approach given by P. A. Zanzotto [20]. Key Words One-dimensional stochastic equations, measurable coefficients, Wiener process, Cauchy process, symmetric stable processes, time change, purely discontinuous processes. MR Subject Classification 60H10, 60J60, 60J65, 60G44.