Approximation of multifractional Brownian motion by absolutely continuous processes. (English. Ukrainian original) Zbl 1237.60029

Ral’chenko, K. V.

Theory Probab. Math. Stat. 82, 115-127 (2011); translation from Teor. Jmovirn. Mat. Stat. No. 82, 115-127.

Multifractional Brownian motion is a generalization of fractional Brownian motion \(B_t^H\) with time-dependent Hurst parameter \(H_t\); in this paper \(H_t \in (\frac12, 1)\) and is a Hölder continuous function of \(t\) with exponent \(\gamma\). The first main result of the paper is an approximation of \(B_t^{H_t}\) by the absolutely continuous average \(B_t^{H_t,\epsilon} := \frac{1}{\phi_t(\epsilon)} \int_t^{t+\phi_t(\epsilon)} B_s^{H_s} ds\), where the length of the interval \(\phi_t(\epsilon)\) satisfies certain regularity conditions. Then \(\| B^{H,\epsilon} - B^H\|_{1,\beta}\) tends to \(0\) in probability as \(\epsilon \to 0^+\), where the norm is a suitable Besov-type norm and \(\beta \in(0, H_{\min})\) is arbitrary; \(H_{\min} = \min\{\gamma, \min_t H_t\}\).
The second main result of the paper is an application of the above result to stochastic differential equations. If a SDE is driven by multifractional Brownian motion, then – under conditions that guarantee the existence and uniqueness of the solution – the approximate solution \(X_t^{\epsilon}\) obtained with the driving process \(B_t^{H_t,\epsilon}\), tends in probability, as \(\epsilon \to 0^+\), to the solution \(X_t\) obtained with the driving process \(B_t^{H_t}\).

Reviewer: Tamas Szabados (Budapest)

MSC:

60G22	Fractional processes, including fractional Brownian motion
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G15	Gaussian processes

Keywords:

multifractional Brownian motion; stochastic differential equations; Gaussian processes

References:

[1]	Robert J. Adler, An introduction to continuity, extrema, and related topics for general Gaussian processes, Institute of Mathematical Statistics Lecture Notes — Monograph Series, vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990. · Zbl 0747.60039
[2]	Albert Benassi, Stéphane Jaffard, and Daniel Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13 (1997), no. 1, 19 – 90 (English, with English and French summaries). · Zbl 0880.60053 · doi:10.4171/RMI/217
[3]	Brahim Boufoussi, Marco Dozzi, and Renaud Marty, Local time and Tanaka formula for a Volterra-type multifractional Gaussian process, Bernoulli 16 (2010), no. 4, 1294 – 1311. · Zbl 1213.60075 · doi:10.3150/10-BEJ261
[4]	Serge Cohen, From self-similarity to local self-similarity: the estimation problem, Fractals: theory and applications in engineering, Springer, London, 1999, pp. 3 – 16. · Zbl 0965.60073
[5]	A. M. Garsia, E. Rodemich, and H. Rumsey Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20 (1970/1971), 565 – 578. · Zbl 0252.60020 · doi:10.1512/iumj.1970.20.20046
[6]	Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. · Zbl 1138.60006
[7]	David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55 – 81. · Zbl 1018.60057
[8]	R. F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, INRIA research report, vol. 2645, 1995.
[9]	K. V. Ral\(^{\prime}\)chenko and G. M. Shevchenko, Properties of the paths of a multifractal Brownian motion, Teor. Ĭmovīr. Mat. Stat. 80 (2009), 106 – 116 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 80 (2010), 119 – 130. · Zbl 1224.60080
[10]	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\(^{\prime}\)skiĭ; Translated from the 1987 Russian original; Revised by the authors.