Numerical proof of existence of fractional powers of Wiener processes

Numerical proof of existence of fractional powers of Wiener processes Motivated by our recent proposal linking stochastic processes and Schrödinger equation, we use the Euler–Maruyama technique to show that a class of Wiener processes exist that are obtained by computing an arbitrary positive power of them. This can be accomplished with a proper set of definitions that makes meaningful the realization at discrete times of these processes and make them computable. Standard results from Itō calculus for integer powers hold as we are just extending them. We provide the results from a Monte Carlo simulation with a large number of samples. We yield evidence for the existence of these processes by recovering from them the standard Brownian motion we started with after power elevation. The perfect coincidence of the numerical results we obtained is a clear evidence of existence of these processes. This could pave the way to a generalization of the concepts of stochastic integral and relative process. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Signal, Image and Video Processing" Springer Journals

Numerical proof of existence of fractional powers of Wiener processes

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Publisher
Springer London
Copyright
Copyright © 2017 by Springer-Verlag London
Subject
Engineering; Signal,Image and Speech Processing; Image Processing and Computer Vision; Computer Imaging, Vision, Pattern Recognition and Graphics; Multimedia Information Systems
ISSN
1863-1703
eISSN
1863-1711
D.O.I.
10.1007/s11760-017-1094-7
Publisher site
See Article on Publisher Site

Abstract

Motivated by our recent proposal linking stochastic processes and Schrödinger equation, we use the Euler–Maruyama technique to show that a class of Wiener processes exist that are obtained by computing an arbitrary positive power of them. This can be accomplished with a proper set of definitions that makes meaningful the realization at discrete times of these processes and make them computable. Standard results from Itō calculus for integer powers hold as we are just extending them. We provide the results from a Monte Carlo simulation with a large number of samples. We yield evidence for the existence of these processes by recovering from them the standard Brownian motion we started with after power elevation. The perfect coincidence of the numerical results we obtained is a clear evidence of existence of these processes. This could pave the way to a generalization of the concepts of stochastic integral and relative process.

Journal

"Signal, Image and Video Processing"Springer Journals

Published: Apr 17, 2017

References

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