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Deterministic characterization of viability for stochastic differential equation driven by fractional Brownian motion∗∗∗

Published online by Cambridge University Press:  22 November 2011

Tianyang Nie  and
Aurel Răşcanu
Tianyang Nie
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China. nietianyang@163.com ; tianyang.nie@uaic.ro Faculty of Mathematics, “Alexandru Ioan Cuza” University, Carol I Blvd, No. 11, 700506 Iasi, Romania; aurel.rascanu@uaic.ro
Aurel Răşcanu
Affiliation:
Faculty of Mathematics, “Alexandru Ioan Cuza” University, Carol I Blvd, No. 11, 700506 Iasi, Romania; aurel.rascanu@uaic.ro “Octav Mayer” Mathematics Institute of the Romanian Academy, Carol I Blvd, No. 8, 700506 Iasi, Romania
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Abstract

In this paper, using direct and inverse images for fractional stochastic tangent sets, we establish the deterministic necessary and sufficient conditions which control that the solution of a given stochastic differential equation driven by the fractional Brownian motion evolves in some particular sets K. As a consequence, a comparison theorem is obtained.

Keywords

Stochastic viabilitystochastic differential equationstochastic tangent setfractional Brownian motion
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 915 - 929
Copyright
© EDP Sciences, SMAI, 2011

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References

J.P., Aubin and G., Da Prato, Stochastic viability and invariance. Ann. Scuola Norm. Super. Pisa Cl. Sci. 27 (1990) 595–694. Google Scholar
F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic calculus for fractional Brownian motion and applications. Springer (2006).
Buckdahn, R., Quincampoix, M. and Rascanu, A., Propriété de viabilité pour des équations différentielles stochastiques rétrogrades et applications à des équations aux derivées partielles. C. R. Acad. Sci. Paris Sér. I 325 (1997) 1159–1162. Google Scholar
Buckdahn, R., Peng, S., Quincampoix, M. and Rainer, C., Existence of stochastic control under state constraints. C. R. Acad. Sci. Paris Sér. I 327 (1998) 17–22. Google Scholar
Buckdahn, R., Quincampoix, M. and Rascanu, A., Viability property for backward stochastic differential equation and applications to partial differential equation. Probab. Theory Relat. Fields 116 (2000) 485–504. Google Scholar
Buckdahn, R., Quincampoix, M., Rainer, C. and Rascanu, A., Viability of moving sets for stochastic differential equation. Adv. Differential Equations 7 (2002) 1045–1072. Google Scholar
Ciotir, I. and Rascanu, A., Viability for stochastic differential equation driven by fractional Brownian motions. J. Differential Equations 247 (2009) 1505–1528. Google Scholar
Mandelbrot, B.B. and Van Ness, J.W., Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422–437. Google Scholar
A., Milian, A note on stochastic invariance for Ito equations. Bull. Pol. Acad. Sci., Math. 41 (1993) 139–150. Google Scholar
Y.S. Mishura, Stochastic calculus for fractional Brownian motion and related processes. Springer (2007).
Nualart, D. and Rascanu, A., Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55–81. Google Scholar
K. Yosida, Functional Analysis. Springer (1971).