Stochastic Calculus with Lévy Processes

In this paper, we construct fractional Lévy processes for any parameter H ∈ (0, 1), as the generalization of the fractional Brownian motion. By using Malliavin calculus, we also define the stochastic integral for fractional Lévy processes.

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Stochastic calculus for convoluted Lévy processes

Bernoulli ◽

10.3150/07-bej115 ◽

2008 ◽

Vol 14 (2) ◽

pp. 499-518 ◽

Cited By ~ 14

Author(s):

Christian Bender ◽

Tina Marquardt

Keyword(s):

Lévy Processes ◽

Levy Processes ◽

Stochastic Calculus

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Lévy Processes and Stochastic Calculus

Journal of the American Statistical Association ◽

10.1198/jasa.2005.s53 ◽

2005 ◽

Vol 100 (472) ◽

pp. 1461-1461

Author(s):

Alexander Novikov

Keyword(s):

Lévy Processes ◽

Levy Processes ◽

Stochastic Calculus

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Local time–space stochastic calculus for Lévy processes

Stochastic Processes and their Applications ◽

10.1016/j.spa.2005.12.002 ◽

2006 ◽

Vol 116 (5) ◽

pp. 757-778 ◽

Cited By ~ 12

Author(s):

Nathalie Eisenbaum

Keyword(s):

Local Time ◽

Lévy Processes ◽

Levy Processes ◽

Stochastic Calculus ◽

Time Space

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Tanaka formula for strictly stable processes

Probability and Mathematical Statistics ◽

10.19195/0208-4147.39.1.3 ◽

2019 ◽

Vol 39 (1) ◽

pp. 39-60

Author(s):

Hiroshi Tsukada

Keyword(s):

Potential Theory ◽

Lévy Processes ◽

Levy Processes ◽

Stochastic Calculus ◽

Local Times ◽

Stable Processes ◽

Tanaka Formula

For symmetric Levy processes, if the local times exist, the Tanaka formula has already been constructed via the techniques in the potential theory by Salminen and Yor 2007. In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index α ∈ 1, 2, including spectrally positive and negative cases in a framework of Ito’s stochastic calculus. Our approach to the existence of local times for such processes is different from that of Bertoin 1996.

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