%0 Journal Article
%T On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes
%A V. P. Kurenok
%J International Journal of Stochastic Analysis
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/258415
%X We consider a one-dimensional stochastic equation , , with respect to a symmetric stable process of index . It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation with respect to the semimartingale and corresponding matrix . In the case of we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. The existence proofs are established using the method of Krylov's estimates for processes satisfying the 2-dimensional equation. On another hand, the Krylov's estimates are based on some analytical facts of independent interest that are also proved in the paper. 1. Introduction Let be a one-dimensional symmetric stable process of index with . In this paper we will study the existence of solutions of the equation where are measurable functions. The existence of solutions is understood in weak sense. In the case of , the coefficients and are assumed to be only measurable satisfying additionally some conditions of boundness. Two important particular cases of (1.1) are the equations If , then is a Brownian motion, and this case has been extensively studied by many authors. The multidimensional analogue of (1.1) with only measurable (instead of continuous) coefficients was first studied by Krylov [1] who proved the existence of solutions assuming the boundness of and and nondegeneracty of . The approach he used was based on -estimates for stochastic integrals of processes satisfying (1.1). Later, the results of Krylov were generalized to the case of nonbounded coefficients in various directions. We mention here only the results of Rozkosz and Slomi¨˝ski [2, 3] who replaced, in particular, the assumption of boundness by the assumption of at most linear growth of the coefficients. The linear growth condition guaranteed the existence of nonexploding solutions. The case of exploding solutions was studied in [4] under assumptions of some local integrability of the coefficients and . In the one-dimensional case with , the results are even stronger. For example, for the time-independent case of the coefficients Engelbert and Schmidt obtained very general existence and uniqueness results in [5]. For the case of the time-independent equation (1.2), one had found even sufficient and necessary conditions for the existence and uniqueness (in general, exploding) solutions [6]. The time-dependent equation (1.2) was studied by several authors; we mention here [2, 7] only. There is less known in the case . The time-independent equation (1.1) with was considered in [8] using the
%U http://www.hindawi.com/journals/ijsa/2012/258415/