Stochastic analysis for vector-valued generalized grey Brownian motion
Authors:
Wolfgang Bock, Martin Grothaus and Karlo Orge
Journal:
Theor. Probability and Math. Statist. 108 (2023), 1-27
MSC (2020):
Primary 46F25, 46F12, 60G22, 33E12, 60H10, 26A33, 60J22
DOI:
https://doi.org/10.1090/tpms/1184
Published electronically:
May 2, 2023
Full-text PDF
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Abstract: In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We characterize these distributions with suitable transformations and give a $d$-dimensional Donsker’s delta function as an example for such distributions. From there, we show the existence of local times and self-intersection local times of vggBm as distributions under some constraints, and compute their corresponding generalized expectations. At the end, we solve a system of linear SDEs driven by a vggBm noise in $d$ dimensions.
References
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Additional Information
Wolfgang Bock
Affiliation:
Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany
Email:
bock@mathematik.uni-kl.de
Martin Grothaus
Affiliation:
Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany
Email:
grothaus@mathematik.uni-kl.de
Karlo Orge
Affiliation:
Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany; and Department of Mathematics and Statistics, Mindanao State University - Iligan Institute of Technology, 9200 Iligan City, Philippines
Email:
orge@mathematik.uni-kl.de, karlo.orge@g.msuiit.edu.ph
Keywords:
Non-Gaussian analysis,
Mittag-Leffler analysis,
generalized functions,
vector-valued generalized grey Brownian motion,
linear stochastic differential equations,
local time
Received by editor(s):
November 17, 2021
Accepted for publication:
June 26, 2022
Published electronically:
May 2, 2023
Additional Notes:
The author K. Orge gratefully acknowledges the financial support by the German Academic Exchange Service (DAAD) in the form of a scholarship on the PhD program “Mathematics in Industry and Commerce” at TU Kaiserslautern.
Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv
