Approximation of a stochastic integral with respect to fractional Brownian motion by integrals with respect to absolutely continuous processes
Author:
T. O. Androshchuk
Translated by:
V. V. Semenov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 73 (2005).
Journal:
Theor. Probability and Math. Statist. 73 (2006), 1929
MSC (2000):
Primary 60H05; Secondary 60G15
Published electronically:
January 17, 2007
MathSciNet review:
2213333
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 1.
Yulīya
Mīshura, An estimate for the ruin probability for models
with longterm dependence, Teor. Ĭmovīr. Mat. Stat.
72 (2005), 93–100 (Ukrainian, with Ukrainian
summary); English transl., Theory Probab. Math. Statist.
72 (2006), 103–111. MR 2168140
(2007b:60164)
 2.
Ilkka
Norros, Esko
Valkeila, and Jorma
Virtamo, An elementary approach to a Girsanov formula and other
analytical results on fractional Brownian motions, Bernoulli
5 (1999), no. 4, 571–587. MR 1704556
(2000f:60053), http://dx.doi.org/10.2307/3318691
 3.
I.
I. \cyr{G}ikhman, I.
I. Gihman, and A.
V. \cyr{S}korokhod, Vvedenie v teoriyu sluchainykh protsessov,
Izdat. “Nauka”, Moscow, 1977 (Russian). Second edition,
revised. MR
0488196 (58 #7758)
 4.
Philip
Protter, Stochastic integration and differential equations,
Applications of Mathematics (New York), vol. 21, SpringerVerlag,
Berlin, 1990. A new approach. MR 1037262
(91i:60148)
 5.
David
Nualart and Aurel
Răşcanu, Differential equations driven by fractional
Brownian motion, Collect. Math. 53 (2002),
no. 1, 55–81. MR 1893308
(2003f:60105)
 6.
M.
Zähle, Integration with respect to fractal functions and
stochastic calculus. I, Probab. Theory Related Fields
111 (1998), no. 3, 333–374. MR 1640795
(99j:60073), http://dx.doi.org/10.1007/s004400050171
 1.
 Yu. S. Mishura, An estimate of ruin probabilities for long range dependence models, Teor. Imovr. Mat. Stat. 72 (2005), 93100; English transl. in Theor. Probability and Math. Statist. 72 (2005), 103111. 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), 571587. 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., PhiladelphiaLondonToronto, 1969. MR 0488196 (58:7758)
 4.
 P. Protter, Stochastic Integration and Differential Equations, SpringerVerlag, 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, 5581. MR 1893308 (2003f:60105)
 6.
 M. Zähle, Integration with respect to fractal functions and stochastic calculus. Part I, Probab. Theory Related Fields 111 (1998), 33372. MR 1640795 (99j:60073)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60H05,
60G15
Retrieve articles in all journals
with MSC (2000):
60H05,
60G15
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:
http://dx.doi.org/10.1090/S0094900007006783
PII:
S 00949000(07)006783
Keywords:
Fractional Brownian motion,
stochastic integral,
convergence of integrals
Received by editor(s):
October 11, 2004
Published electronically:
January 17, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
