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.
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Notes
A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q. \(C\ell (V,Q)\) has the condition \(v^2 = Q(v)1\) for all \(v\in V\). See https://en.wikipedia.org/wiki/Clifford_algebra.
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Frasca, M., Farina, A. Numerical proof of existence of fractional powers of Wiener processes. SIViP 11, 1365–1370 (2017). https://doi.org/10.1007/s11760-017-1094-7
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DOI: https://doi.org/10.1007/s11760-017-1094-7