Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 6 (2001) > Paper 7 open journal systems 


Finitely Polynomially Determined Lévy Processes

Arindam Sengupta, Indian Statistical Institute
Anish Sarkar, Indian Statistical Institute (Delhi Centre)


Abstract
A time-space harmonic polynomial for a continuous-time process $X={X_t : t ge 0} $ is a two-variable polynomial $ P $ such that $ { P,(t,X_t) : t ge 0 } $ is a martingale for the natural filtration of $ X $. Motivated by Lévy's characterisation of Brownian motion and Watanabe's characterisation of the Poisson process, we look for classes of processes with reasonably general path properties in which a characterisation of those members whose laws are determined by a finite number of such polynomials is available. We exhibit two classes of processes, the first containing the Lévy processes, and the second a more general class of additive processes, with this property and describe the respective characterisations.


Full text: PDF

Pages: 1-22

Published on: August 30, 2000
Bibliography
  1. P. Billingsley. Probability and Measure. John Wiley, New York, 1986. Math. Review 87f:60001
  2. P. Billingsley. Convergence of Probability Measures. John Wiley, London, 1968. Math. Review 38:1718
  3. J. L. Doob. Stochastic Processes. John Wiley & Sons, New York, 1952. Math. Review 15:445b
  4. W. Feller. An Introduction to Probability Theory and Its Applications. Wiley Eastern University Edition, 1988. Math. Review 19:466
  5. B. V. Gnedenko. The Theory of Probability. Chelsea, New York, 1963. Math. Review 36:912
  6. B. V. Gnedenko and A. N. Kolmogorov. Limit Distributions for Sums of Independent Random Variables. Addison-Wesley Publications, Cambridge, 1954. Math. Review 16:52d
  7. K. Ito. Lectures on Stochastic Processes. Tata Institute of Fundamental Research Lecture Notes. Narosa, New Delhi, 1984. Math. Review 86f:60049
  8. A. S. Kechris. Classical Descriptive Set Theory. Springer-Verlag, Graduate Texts in Mathematics # 156, 1995. Math. Review 96e:03057
  9. P. McGill, B. Rajeev and B. V. Rao. Extending Lévy's characterisation of Brownian motion. Seminaire de Prob. XXIII , Lecture Notes in Mathematics # 1321, 163 - 165, Springer-Verlag, 1988. Math. Review 89g:60252
  10. Arindam Sengupta. Time-space Harmonic Polynomials for Stochastic Processes. Ph. D. Thesis (unpublished), Indian Statistical Institute, Calcutta, 1998. Math. Review number not available.
  11. Arindam Sengupta. Time-space Harmonic Polynomials for Continuous-time Processes and an Extension. Journal of Theoretical Probability, 13 , 951 - 976, 2000. Math. Review number not available.
  12. S. Watanabe. On discontinuous additive functionals and Lévy measures of a Markov process. Japanese J. Math. , 34 , 53 -- 70, 1964. Math. Review 32:3137
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489