The Annals of Probability

On stochastic differential equations driven by a Cauchy process and other stable Lévy motions

Pio Andrea Zanzotto

Full-text: Open access

Abstract

We consider the class of one-dimensional stochastic differential equations

$$dX_t = b(X_{t-})dZ_t, \quad t \geq 0,$$

where $b$ is a Borel measurable real function and $Z$ is a strictly $\alpha$-stable Lévy process $(0 < \alpha \leq 2)$. Weak solutions are investigated improving previous results of the author in various ways.

In particular, for the equation driven by a strictly 1-stable Lévy process, a sufficient existence condition is proven.

Also we extend the weak existence and uniqueness exact criteria due to Engelbert and Schmidt for the Brownian case (i.e., $\alpha = 2$) to the class of equations with $\alpha$ such that $1 < \alpha \leq 2$. The results employ some representation properties with respect to strictly stable Lévy processes.

Article information

Source
Ann. Probab. Volume 30, Number 2 (2002), 802-825.

Dates
First available: 7 June 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1023481008

Mathematical Reviews number (MathSciNet)
MR1905857

Digital Object Identifier
doi:10.1214/aop/1023481008

Zentralblatt MATH identifier
1017.60058

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J30

Keywords
strictly $\alpha$-stable Lévy processes Cauchy process stochastic differential equations weak existence "local" existence time change quadratic pure-jump semimartingales representation stable integrals

Citation

Zanzotto, Pio Andrea. On stochastic differential equations driven by a Cauchy process and other stable Lévy motions. The Annals of Probability 30 (2002), no. 2, 802--825. doi:10.1214/aop/1023481008. http://projecteuclid.org/euclid.aop/1023481008.


Export citation

References

  • [1] BERTOIN, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • [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.
  • [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.
  • [5] IKEDA, N. and WATANABE, SH. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
  • [6] JACOD, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • [7] JACOD, J. and SHIRYAEV, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • [8] KALLENBERG, O. (1992). Some time change representations of stable integrals, via predictable transformations of local martingales. Stochastic Process. Appl. 40 199-223.
  • [9] KALLENBERG, O. (1997). Foundations of Modern Probability. Springer, New York.
  • [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.
  • [12] PROTTER, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.
  • [13] REVUZ, D. and YOR, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • [14] SAMORODNITSKY, G. and TAQQU, M. S. (1994). Stable Non-Gaussian Random ProcessesStochastic Models with Infinite Variance. Chapman and Hall, London.
  • [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.
  • [16] STONE, CH. (1963). The set of zeros of a semistable process. Illinois J. Math. 7 631-637.
  • [17] TAYLOR, S. J. (1967). Sample path properties of a transient stable process. J. Math. Mech. 16 1229-1246.
  • [18] ZANZOTTO, P. A. (1997). On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion. Stochastic Process. Appl. 68 209-228.
  • [19] ZANZOTTO, P. A. (1998). Representation of a class of semimartingales as stable integrals. Teor. Veroyatnost. i Primenen. 43 808-818. (In English.)
  • [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.
  • VIA TOMADINI, 301A 33100 UDINE ITALY E-MAIL: zanzotto@dm.unipi.it