Electronic Journal of Probability

Finitely Polynomially Determined Lévy Processes

Arindam Sengupta and Anish Sarkar

Full-text: Open access

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.

Article information

Source
Electron. J. Probab., Volume 6 (2001), paper no. 7, 22 pp.

Dates
Accepted: 30 August 2000
First available in Project Euclid: 19 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1461097637

Digital Object Identifier
doi:10.1214/EJP.v6-80

Mathematical Reviews number (MathSciNet)
MR1831802

Zentralblatt MATH identifier
0974.60026

Subjects
Primary: 60G44: Martingales with continuous parameter
Secondary: 60J30

Keywords
Lévy process additive process Lévy's characterisation Lévy measure Kolmogorov measure

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Sengupta, Arindam; Sarkar, Anish. Finitely Polynomially Determined Lévy Processes. Electron. J. Probab. 6 (2001), paper no. 7, 22 pp. doi:10.1214/EJP.v6-80. https://projecteuclid.org/euclid.ejp/1461097637


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References

  • P. Billingsley. Probability and Measure. John Wiley, New York, 1986.
  • P. Billingsley. Convergence of Probability Measures. John Wiley, London, 1968.
  • J. L. Doob. Stochastic Processes. John Wiley & Sons, New York, 1952.
  • W. Feller. An Introduction to Probability Theory and Its Applications. Wiley Eastern University Edition, 1988.
  • B. V. Gnedenko. The Theory of Probability. Chelsea, New York, 1963.
  • B. V. Gnedenko and A. N. Kolmogorov. Limit Distributions for Sums of Independent Random Variables. Addison-Wesley Publications, Cambridge, 1954.
  • K. Ito. Lectures on Stochastic Processes. Tata Institute of Fundamental Research Lecture Notes. Narosa, New Delhi, 1984.
  • A. S. Kechris. Classical Descriptive Set Theory. Springer-Verlag, Graduate Texts in Mathematics # 156, 1995.
  • 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.
  • Arindam Sengupta. Time-space Harmonic Polynomials for Stochastic Processes. Ph. D. Thesis (unpublished), Indian Statistical Institute, Calcutta, 1998.
  • Arindam Sengupta. Time-space Harmonic Polynomials for Continuous-time Processes and an Extension. Journal of Theoretical Probability, 13, 951 - 976, 2000.
  • S. Watanabe. On discontinuous additive functionals and Lévy measures of a Markov process. Japanese J. Math., 34, 53 – 70, 1964.