Support of a Marcus equation in Dimension 1

Thomas Simon (Humboldt-Universitat zu Berlin)

Abstract


The purpose of this note is to give a support theorem in the Skorohod space for a one-dimensional Marcus differential equation driven by a Lévy process, without any assumption on the latter. We also give a criterion ensuring that the support of the equation is the whole Skorohod space. This improves, in dimension 1, a result of H. Kunita.

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Pages: 149-157

Publication Date: September 7, 2000

DOI: 10.1214/ECP.v5-1028

References

  • Applebaum, David; Kunita, Hiroshi. Lévy flows on manifolds and Lévy processes on Lie groups. J. Math. Kyoto Univ. 33 (1993), no. 4, 1103--1123. MR1251218
  • Doss, Halim. Liens entre équations différentielles stochastiques et ordinaires. (French) Ann. Inst. H. Poincaré Sect. B (N.S.) 13 (1977), no. 2, 99--125. MR0451404
  • M. Errami, F. Russo and P. Vallois. Itô formula for CC^1,λ functions of a càdlàg process and related calculus. Prépublication de l'Institut Élie Cartan, Nancy, 1999.
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 MR1011252
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1 MR0959133
  • Kunita, Hiroshi. Stochastic flows with self-similar properties. Stochastic analysis and applications (Powys, 1995), 286--300, World Sci. Publ., River Edge, NJ, 1996. MR1453139
  • Kunita, Hiroshi. Canonical stochastic differential equations based on Lévy processes and their supports. Stochastic dynamics (Bremen, 1997), 283--304, Springer, New York, 1999. MR1678491
  • H. Kunita and J. P. Oh. Existence of density for canonical differential equations with jumps. Preprint, 1999.
  • Kurtz, Thomas G.; Pardoux, Étienne; Protter, Philip. Stratonovich stochastic differential equations driven by general semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 2, 351--377. MR1324812
  • Marcus, Steven I. Modeling and analysis of stochastic differential equations driven by point processes. IEEE Trans. Inform. Theory 24 (1978), no. 2, 164--172. MR0487784
  • Marcus, Steven I. Modeling and approximation of stochastic differential equations driven by semimartingales. Stochastics 4 (1980/81), no. 3, 223--245. MR0605630
  • Simon, Thomas. Support theorem for jump processes. Stochastic Process. Appl. 89 (2000), no. 1, 1--30. MR1775224
  • Th. Simon. Sur les petites déviations d'un processus de Lévy. To appear in Potential Analysis.
  • Th. Simon. Über eine nichtlineare Itô-Differentialgleichung. In preparation.
  • Stroock, Daniel W.; Varadhan, S. R. S. On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 333--359. Univ. California Press, Berkeley, Calif., 1972. MR0400425
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