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Drifted Brownian motions governed by fractional tempered derivatives
Volume 5, Issue 4 (2018), pp. 445–456
Mirko D’Ovidio   Francesco Iafrate   Enzo Orsingher  

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https://doi.org/10.15559/18-VMSTA114
Pub. online: 19 September 2018      Type: Research Article      Open accessOpen Access

Received
24 April 2018
Revised
28 June 2018
Accepted
29 August 2018
Published
19 September 2018

Abstract

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann–Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.

References

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Beghin, L.: On fractional tempered stable processes and their governing differential equations. J. Comput. Phys. 1, 29–39 (2015). MR3342454. https://doi.org/10.1016/j.jcp.2014.05.026
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D’Ovidio, M.: On the fractional counterpart of the higher-order equations. Stat. Probab. Lett. 81, 1929–1939 (2011). MR2845910. https://doi.org/10.1016/j.spl.2011.08.004
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D’Ovidio, M., Orsingher, E.: Bessel processes and hyperbolic Brownian motions stopped at different random times. Stoch. Process. Appl. 121, 441–465 (2011). MR2763091. https://doi.org/10.1016/j.spa.2010.11.002
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D’Ovidio, M., Orsingher, E., Toaldo, B.: Time changed processes governed by space-time fractional telegraph equations. Stoch. Anal. Appl. 32, 1009–1045 (2014). MR3270693. https://doi.org/10.1080/07362994.2014.962046
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Gradshteyn, I., Ryzhik, I.: Table of Integrals, Series, and Products. Academic Press, New York (1999). MR0669666
[6] 
Iafrate, F., Orsingher, E.: Last zero crossing of an iterated Brownian motion with drift. https://arxiv.org/abs/1803.00877. Accessed 2 Mar 2018.
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Sabzikar, F., Meerschaert, M.M., Chen, J.: Tempered fractional calculus. J. Comput. Phys. 121, 14–28 (2015). MR3342453. https://doi.org/10.1016/j.jcp.2014.04.024

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Keywords
Tempered fractional derivatives drifted Brownian motion

MSC2010
34A08 60J65

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