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

2012/06/15 by Doering, Leif · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1206.3515

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.

Cited by

• On the passage times of self-similar Gaussian processes on curved boundaries

Related

• A Jump Type SDE Approach to Positive Self-Similar Markov Processes
• Exit problems for positive self-similar Markov processes with one-sided jumps
• Hitting properties and non-uniqueness for SDE driven by stable processes
• Real Self-Similar Processes Started from the Origin
• Self-similar branching Markov chains