Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)
Entropy minimization for a mixture of standard and fractional Brownian motions
V. I. Makogin, Yu. S. Mishura, H. S. Zhelezniak
Abstract: In this paper, we consider an entropy-type functional for the sum of the Wiener process and the fractional Brownian motion with a trend. The solution of the minimization problem of such a functional in the space of $ L_2 $ -functions is found. The properties of the solution norm are investigated, and also the variant of the minimization problem on the space of constant functions is considered. As a result of the proved continuity of weighted integral Riemann-Liouville operators, $ L_2 $ -continuity of the minimization problem solution as a function of the Hurst index is shown.
1. H. Follmer, A. Schied, Stochastic nance: an introduction in discrete time, Walter de Gruyter, Berlin, 2002.
2. G. G. Judge, R. C. Mittelhammer, An Information Theoretic Approach to Econometrics, Cambridge University Press, Cambridge, 2012.
3. Y. Mishura, H. Zhelezniak, Extreme measures for entropy functionals, Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics & Mathematics, (2017), no. 4, 15-20.
4. Y. S. Mishura, H. S. Zhelezniak, Calculation of extremums of entropy functionals, Theor. Probability and Math. Statist., 99 (2019), 177-186.
5. M. Avellaneda, C. Friedman, R. Holmes, D. Samperi, Calibrating volatility surfaces via relative entropy minimization, Applied Mathematical Finance, (1997), 7-64.
6. C. Leonard, Minimization of entropy functionals, Journal of Mathematical Analysis and Applications, Elsevier, 346 (2008), no. 1, 183-204.
7. C. Jost, Transformation formulas for fractional Brownian motion, Stochastic Processes and their Applications, 116 (2006), no. 10, 1341-1357.
8. A. MacKay, A. Melnikov, Y. Mishura, Optimization of small deviation for mixed fractional Brownian motion with trend, Stochastics, 90 (2018), no. 7, 1087-1110.
9. Yu. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Springer Science & Business Media, 2008.
10. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications., Gordon and Breach, Yverdon, 1993.