On stochastic differential equations driven by a Cauchy process and other stable Lévy motions. (English) Zbl 1017.60058

The author considers existence and uniqueness of weak solutions of the one-dimensional stochastic differential equation \[ X_t = X_0 + \int_{]0,t]} b(X_{s-}) dZ_s, \quad t \geq 0, \] where \(X_0\) has an arbitrary distribution, \(b\) is a Borel measurable real function, \(Z\) is a strictly \(\alpha\)-stable Lévy process, \(0 < \alpha \leq 2\), \(X_0\) and \(Z\) are stochastically independent, and (when \(\alpha < 2\)) either \(Z\) is symmetric or \(b\) is nonnegative. The weak existence and uniqueness exact criterion of Engelbert and Schmidt for \(Z\) Brownian motion, i.e.\(\alpha=2\), is extended to \(\alpha\)-stable processes with \(1 < \alpha \leq 2\). This criterion is given in terms of the zero set and the \(\alpha\)-singularity set for the coefficient \(b\). Sufficient conditions based only on the coefficient \(b\) are given for the existence of a solution in the cases that \(Z\) is a Cauchy process, that \(b\) is nonnegative and \(Z\) is an \(1\)-stable Lévy process, and that \(Z\) is an \(\alpha\)-stable Lévy process with \(0 < \alpha < 1\). A local version (LH)(x) of the condition (H)(x) in the paper of the author [Stochastic Processes Appl. 68, No. 2, 209-228 (1997; Zbl 0911.60037)] is introduced, and the meaning of the condition (H)(x) is clarified. Sufficient conditions for the existence of nontrivial local weak solutions starting from a fixed point \(X_0 \equiv x\) are given in terms of the condition (LH)(x). The proofs rely on a representation property with respect to strictly stable Lévy processes.


60G52 Stable stochastic processes
60G51 Processes with independent increments; Lévy processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)


Zbl 0911.60037
Full Text: DOI


[1] BERTOIN, J. (1996). Lévy Processes. Cambridge Univ. Press. · Zbl 0861.60003
[2] DELLACHERIE, C. (1971). Capacités et Processus Stochastiques. Springer, Berlin.
[3] ENGELBERT, H. J. and SCHMIDT, W. (1985). On solutions of one-dimensional stochastic differential equations without drift.Wahrsch. Verw. Gebiete 68 287-314. · Zbl 0535.60049
[4] ENGELBERT, H. J. and SCHMIDT, W. (1985). On one-dimensional stochastic differential equations with generalized drift. Lecture Notes in Control and Information Sciences 69 143-155. Springer, Berlin. · Zbl 0583.60052
[5] IKEDA, N. and WATANABE, SH. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. · Zbl 0495.60005
[6] JACOD, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Springer, Berlin. · Zbl 0414.60053
[7] JACOD, J. and SHIRYAEV, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin. · Zbl 0635.60021
[8] KALLENBERG, O. (1992). Some time change representations of stable integrals, via predictable transformations of local martingales. Stochastic Process. Appl. 40 199-223. · Zbl 0754.60044
[9] KALLENBERG, O. (1997). Foundations of Modern Probability. Springer, New York. · Zbl 0892.60001
[10] KARATZAS, I. and SHREVE, S. E. (1994). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
[11] PORT, S. C. (1967). Hitting times for transient stable processes. Pacific J. Math. 21 161-165. · Zbl 0154.18904
[12] PROTTER, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin. · Zbl 0694.60047
[13] REVUZ, D. and YOR, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin. · Zbl 0731.60002
[14] SAMORODNITSKY, G. and TAQQU, M. S. (1994). Stable Non-Gaussian Random ProcessesStochastic Models with Infinite Variance. Chapman and Hall, London. · Zbl 0925.60027
[15] SATO, K. (1997). Time evolution of Lévy processes. In Trends in Probability and Related Analysis (N. Kono and N.-R. Shieh, eds.) 35-82. World Scientific, Singapore. · Zbl 1010.60042
[16] STONE, CH. (1963). The set of zeros of a semistable process. Illinois J. Math. 7 631-637. · Zbl 0121.12906
[17] TAYLOR, S. J. (1967). Sample path properties of a transient stable process. J. Math. Mech. 16 1229-1246. · Zbl 0178.19301
[18] ZANZOTTO, P. A. (1997). On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion. Stochastic Process. Appl. 68 209-228. · Zbl 0911.60037
[19] ZANZOTTO, P. A. (1998). Representation of a class of semimartingales as stable integrals. Teor. Veroyatnost. i Primenen. 43 808-818. (In English.) · Zbl 0956.60055
[20] ZANZOTTO, P. A. (1998). On stochastic differential equations driven by Cauchy process and the other stable Lévy motions. Technical Report 2.312.1127, Dept. Mathematics, Univ. Pisa.
[21] ZANZOTTO, P. A. (1999). On stochastic differential equations driven by Cauchy process and the other -stable motions. In Mini-Proceedings: Conference on Lévy Processes: Theory and Applications 179-183. MaPhySto, Univ. Aarhus.
[22] VIA TOMADINI, 301A 33100 UDINE ITALY E-MAIL: zanzotto@dm.unipi.it
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