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 Slomiński [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
References
[1]
N. V. Krylov, Controlled Diffusion Processes, vol. 14 of Applications of Mathematics, Springer, New York, NY, USA, 1980.
[2]
A. Rozkosz and L. S？omiński, “On weak solutions of one-dimensional SDEs with time-dependent coefficients,” Stochastics and Stochastics Reports, vol. 42, pp. 199–208, 1993.
[3]
A. Rozkosz and L. S？omiński, “On existence and stability of weak solutions of multidimensional stochastic differential equations with measurable coefficients,” Stochastic Processes and their Applications, vol. 37, no. 2, pp. 187–197, 1991.
[4]
V. P. Kurenok and A. N. Lepeyev, “On multi-dimensional SDEs with locally integrable coefficients,” The Rocky Mountain Journal of Mathematics, vol. 38, no. 1, pp. 139–174, 2008.
[5]
H. J. Engelbert and W. Schmidt, “On one-dimensional stochastic differential equations with generalized drift,” in Stochastic Differential Systems, vol. 69 of Lecture Notes in Control and Information Science, pp. 143–155, Springer, Berlin, Germany, 1985.
[6]
H. J. Engelbert and W. Schmidt, “Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations III,” Mathematische Nachrichten, vol. 151, pp. 149–197, 1991.
[7]
T. Senf, “On one-dimensional stochastic differential equations without drift and with time-dependent diffusion coefficients,” Stochastics and Stochastics Reports, vol. 43, no. 3-4, pp. 199–220, 1993.
[8]
V. Kurenok, “A note on -estimates for stable integrals with drift,” Transactions of the American Mathematical Society, vol. 360, no. 2, pp. 925–938, 2008.
[9]
P. A. Zanzotto, “On stochastic differential equations driven by a Cauchy process and other stable Lévy motions,” The Annals of Probability, vol. 30, no. 2, pp. 802–825, 2002.
[10]
G. Pragarauskas and P. A. Zanzotto, “On one-dimensional stochastic differential equations with respect to stable processes,” Matematikos ir Informatikos Institutas, vol. 40, no. 3, pp. 361–385, 2000.
[11]
H.-J. Engelbert and V. P. Kurenok, “On one-dimensional stochastic equations driven by symmetric stable processes,” in Stochastic Processes and Related Topics, R. Buckdahn, H. J. Engelbert, and M. Yor, Eds., vol. 12, pp. 81–109, Taylor & Francis, London, UK, 2002.
[12]
H. Tanaka, M. Tsuchiya, and S. Watanabe, “Perturbation of drift-type for Lévy processes,” Journal of Mathematics of Kyoto University, vol. 14, pp. 73–92, 1974.
[13]
N. I. Portenko, “Some perturbations of drift-type for symmetric stable processes,” Random Operators and Stochastic Equations, vol. 2, no. 3, pp. 211–224, 1994.
[14]
V. P. Kurenok, “Stochastic equations with time-dependent drift driven by Levy processes,” Journal of Theoretical Probability, vol. 20, no. 4, pp. 859–869, 2007.
[15]
O. Kallenberg, Foundations of Modern Probability, Springer, New York, NY, USA, 1997.
[16]
J. Jacod, Calcul Stochastique et Problèmes de Martingales, vol. 714 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1979.
[17]
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, vol. 293, Springer, Berlin, Germany, 3rd edition, 1999.
[18]
J. Rosiński and W. A. Woyczyński, “On It？ stochastic integration with respect to p-stable motion: inner clock, integrability of sample paths, double and multiple integrals,” The Annals of Probability, vol. 14, no. 1, pp. 271–286, 1986.
[19]
C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel. Chapitres V à VIII, vol. 1385, Hermann, Paris, France, 1980.
[20]
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, vol. 24, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1989.
[21]
D. Aldous, “Stopping times and tightness,” Annals of Probability, vol. 6, no. 2, pp. 335–340, 1978.
