Electronic Journal of Probability

Finitely Polynomially Determined Lévy Processes

Arindam Sengupta and Anish Sarkar

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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