Deterministic characterization of viability for stochastic differential equation driven by fractional brownian motion
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 915-929.

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.

DOI : 10.1051/cocv/2011188
Classification : 60H10, 60H20, 60G22
Mots clés : stochastic viability, stochastic differential equation, stochastic tangent set, fractional brownian motion
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     author = {Nie, Tianyang and R\u{a}\c{s}canu, Aurel},
     title = {Deterministic characterization of viability for stochastic differential equation driven by fractional brownian motion},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {915--929},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {4},
     year = {2012},
     doi = {10.1051/cocv/2011188},
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     zbl = {1263.60052},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2011188/}
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Nie, Tianyang; Răşcanu, Aurel. Deterministic characterization of viability for stochastic differential equation driven by fractional brownian motion. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 915-929. doi : 10.1051/cocv/2011188. http://www.numdam.org/articles/10.1051/cocv/2011188/

[1] J.P. Aubin and G. Da Prato, Stochastic viability and invariance. Ann. Scuola Norm. Super. Pisa Cl. Sci. 27 (1990) 595-694. | Numdam | MR | Zbl

[2] F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic calculus for fractional Brownian motion and applications. Springer (2006). | Zbl

[3] R. Buckdahn, M. Quincampoix and A. Rascanu, 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. | MR | Zbl

[4] R. Buckdahn, S. Peng, M. Quincampoix and C. Rainer, Existence of stochastic control under state constraints. C. R. Acad. Sci. Paris Sér. I 327 (1998) 17-22. | MR | Zbl

[5] R. Buckdahn, M. Quincampoix and A. Rascanu, Viability property for backward stochastic differential equation and applications to partial differential equation. Probab. Theory Relat. Fields 116 (2000) 485-504. | MR | Zbl

[6] R. Buckdahn, M. Quincampoix, C. Rainer and A. Rascanu, Viability of moving sets for stochastic differential equation. Adv. Differential Equations 7 (2002) 1045-1072. | MR | Zbl

[7] I. Ciotir and A. Rascanu, Viability for stochastic differential equation driven by fractional Brownian motions. J. Differential Equations 247 (2009) 1505-1528. | MR | Zbl

[8] B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422-437. | MR | Zbl

[9] A. Milian, A note on stochastic invariance for Ito equations. Bull. Pol. Acad. Sci., Math. 41 (1993) 139-150. | MR | Zbl

[10] Y.S. Mishura, Stochastic calculus for fractional Brownian motion and related processes. Springer (2007). | MR | Zbl

[11] D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. | MR | Zbl

[12] K. Yosida, Functional Analysis. Springer (1971).

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