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Approximation of a stochastic integral with respect to fractional Brownian motion by integrals with respect to absolutely continuous processes
Author(s):
T.
O.
Androshchuk
Translated by:
V. V. Semenov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 73
(2005).
Journal:
Theor. Probability and Math. Statist.
No. 73
(2006),
19-29.
MSC (2000):
Primary 60H05;
Secondary 60G15
Posted:
January 17, 2007
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Abstract:
We consider an absolutely continuous process converging in the mean square sense to a fractional Brownian motion. We obtain sufficient conditions that the integral with respect to this process converges to the integral with respect to the fractional Brownian motion.
References:
-
- 1.
- Yu. S. Mishura, An estimate of ruin probabilities for long range dependence models, Teor. Imov
 r. Mat. Stat. 72 (2005), 93-100; English transl. in Theor. Probability and Math. Statist. 72 (2005), 103-111. MR 2168140 - 2.
- I. Norros, E. Valkeila, and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 55 (1999), 571-587. MR 1704556 (2000f:60053)
- 3.
- I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, Second edition, ``Nauka'', Moscow, 1977; English transl. of the first edition, Scripta Technica, Inc. W. B. Saunders Co., Philadelphia-London-Toronto, 1969. MR 0488196 (58:7758)
- 4.
- P. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, New York, 1990. MR 1037262 (91i:60148)
- 5.
- D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Mat. 53 (2002), no. 1, 55-81. MR 1893308 (2003f:60105)
- 6.
- M. Zähle, Integration with respect to fractal functions and stochastic calculus. Part I, Probab. Theory Related Fields 111 (1998), 33-372. MR 1640795 (99j:60073)
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Additional Information:
T.
O.
Androshchuk
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
nutaras@univ.kiev.ua
DOI:
10.1090/S0094-9000-07-00678-3
PII:
S 0094-9000(07)00678-3
Keywords:
Fractional Brownian motion,
stochastic integral,
convergence of integrals
Received by editor(s):
11/OCT/2004
Posted:
January 17, 2007
Copyright of article:
Copyright
2007,
American Mathematical Society
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