Electronic Journal of Probability

The Symbol Associated with the Solution of a Stochastic Differential Equation

Rene Schilling and Alexander Schnurr

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We consider stochastic differential equations which are driven by multidimensional Levy processes. We show that the infinitesimal generator of the solution is a pseudo-differential operator whose symbol is calculated explicitely. For a large class of Feller processes many properties of the sample paths can be derived by analysing the symbol. It turns out that the solution of the SDE under consideration is a Feller process if the coefficient of the SDE is bounded and that the symbol is of a particulary nice structure.

Article information

Electron. J. Probab. Volume 15 (2010), paper no. 43, 1369-1393.

Accepted: 18 September 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Primary: 60J75: Jump processes
Secondary: 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx] 60H20: Stochastic integral equations 60J25: Continuous-time Markov processes on general state spaces 60G51: Processes with independent increments; Lévy processes 60G17: Sample path properties

stochastic differential equation L'evy process semimartingale pseudo-differential operator Blumenthal-Getoor index sample path properties

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Schilling, Rene; Schnurr, Alexander. The Symbol Associated with the Solution of a Stochastic Differential Equation. Electron. J. Probab. 15 (2010), paper no. 43, 1369--1393. doi:10.1214/EJP.v15-807. http://projecteuclid.org/euclid.ejp/1464819828.

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