On driftless one-dimensional SDE’s with respect to stable Levy processes

Abstract

The time-dependent SDE dX t = b(t, X t−)dZ t with X 0 = x 0 ∈ ℝ, and a symmetric α-stable process Z, 1 < α ⩽ 2, is considered. We study the existence of nonexploding solutions of the given equation through the existence of solutions of the equation $$dA_t = \left| b \right|^\alpha (t,\bar Z \circ A_t )dt$$ in class of time change processes, where $$\bar Z$$ is a symmetric stable process of the same index α as Z. The approach is based on using the time change method, Krylov’s estimates for stable integrals, and properties of monotone convergence. The main existence result extends the results of Pragarauskas and Zanzotto (2000) for 1 < α < 2 and those of T. Senf (1993) for α = 2.