|
|
|---|
ISSN 1547-7363(online) ISSN 0094-9000(print)
Minimization of the entropy for a mixture of standard and fractional Brownian motions
Authors:
V. I. Makogin, Yu. S. Mishura and G. S. Zheleznyak
Journal:
Theor. Probability and Math. Statist. 101 (2020), 193-215
MSC (2020):
Primary 60G22, 60J65; Secondary 94A17
DOI:
https://doi.org/10.1090/tpms/1121
Published electronically:
January 5, 2021
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: An entropy-type functional for the sum of a Wiener process and a fractional Brownian motion with a drift is considered in this paper. A solution of the problem of minimization of such a functional is found in the space of $L_2$ functions. Properties of the norm of the solution are investigated and a version of the problem of minimization is considered in the space of constant functions. The $L_2$ continuity of the solution of minimization problem with respect to the Hurst index is shown as a corollary of the continuity of weighted Riemann–Liouville integral operators proved in the paper.
- Hans Föllmer and Alexander Schied, Stochastic finance, De Gruyter Studies in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 2002. An introduction in discrete time. MR 1925197
- George G. Judge and Ron C. Mittelhammer, An information theoretic approach to econometrics, Cambridge University Press, Cambridge, 2012. MR 3135258
- Y. Mishura and H. Zhelezniak, Extreme measures for entropy functionals, Bull. Nat. Univ. Kyiv. Series: Physics & Mathematics (2017), no. 4, 15–20.
- Yu. S. Mīshura and G. S. Zheleznyak, Determining the extrema of entropy functionals, Teor. Ĭmovīr. Mat. Stat. 99 (2018), 177–186 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 99 (2019), 199–210. MR 3908666, DOI https://doi.org/10.1090/tpms/1090
- M. Avellaneda, C. Friedman, R. Holmes, and D. Samperi, Calibrating volatility surfaces via relative entropy minimization, Appl. Math. Finance (1997), 7–64.
- Christian Léonard, Minimization of entropy functionals, J. Math. Anal. Appl. 346 (2008), no. 1, 183–204. MR 2428283, DOI https://doi.org/10.1016/j.jmaa.2008.04.048
- Céline Jost, Transformation formulas for fractional Brownian motion, Stochastic Process. Appl. 116 (2006), no. 10, 1341–1357. MR 2260738, DOI https://doi.org/10.1016/j.spa.2006.02.006
- Anne MacKay, Alexander Melnikov, and Yuliya Mishura, Optimization of small deviation for mixed fractional Brownian motion with trend, Stochastics 90 (2018), no. 7, 1087–1110. MR 3854528, DOI https://doi.org/10.1080/17442508.2018.1478835
- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2020): 60G22, 60J65, 94A17
Retrieve articles in all journals with MSC (2020): 60G22, 60J65, 94A17
Additional Information
V. I. Makogin
Affiliation:
Institute of Stochastics, Ulm University, Ulm 89069, Germany
Email:
vitalii.makogin@uni-ulm.de
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
myus@univ.kiev.ua
G. S. Zheleznyak
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
hanna.zhelezniak@gmail.com
Keywords:
Wiener process,
fractional Brownian motion,
Radon–Nikodym derivative,
entropy functional,
minimization,
maximization
Received by editor(s):
June 12, 2019
Published electronically:
January 5, 2021
Article copyright:
© Copyright 2020
American Mathematical Society