A Jump-Type SDE Approach to Real-Valued Self-Similar Markov Processes

Abstract : In his 1972 paper, John Lamperti characterized all positive self-similar Markov processes as time-changes of exponentials of Levy processes. In the past decade the problem of classifying all non-negative self-similar Markov processes that do not necessarily have zero as a trap has been solved gradually via connections to ladder height processes and excursion theory. Motivated by a recent article of Chaumont, Rivero, Panti we classify via jump-type SDEs the symmetric real-valued self-similar Markov processes that only decrease the absolute value by jumps and leave zero continuously. Our construction of these self-similar processes involves a pseudo excursion construction and singular stochastic calculus arguments ensuring that solutions to the SDEs spend zero time at zero to avoid problems caused by a "bang-bang" drift.
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Contributor : Leif Doering <>
Submitted on : Monday, June 18, 2012 - 10:38:51 AM
Last modification on : Sunday, March 31, 2019 - 1:16:41 AM

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  • HAL Id : hal-00709194, version 1
  • ARXIV : 1206.3515


Leif Doering. A Jump-Type SDE Approach to Real-Valued Self-Similar Markov Processes. 2012. ⟨hal-00709194⟩



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