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Journal of Theoretical Probability

, Volume 32, Issue 1, pp 395–416 | Cite as

Viability for Stochastic Differential Equations Driven by G-Brownian Motion

  • Peng Luo
  • Falei WangEmail author
Article
First Online:
  • 219 Downloads

Abstract

In this paper, we prove a type of Nagumo theorem on viability properties for stochastic differential equations driven by G-Brownian motion (G-SDEs). In particular, an equivalent criterion is formulated through stochastic contingent and tangent sets. Moreover, by the approach of direct and inverse images for stochastic tangent sets we present checkable conditions which keep the solution of a given G-SDE evolving in some particular sets.

Keywords

Stochastic viability Stochastic differential equation Stochastic tangent set G-Brownian motion 

Mathematics Subject Classification (2010)

60H30 60H10 
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Notes

Acknowledgements

The authors would like to thank the editor and the anonymous referee for their helpful discussions and suggestions.

References

  1. 1.
    Aubin, J.P., Da Prato, G.: Stochastic viability and invariance. Ann. Sc. Norm. Super. Pisa Cl. Sci. 27, 595–694 (1990)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aubin, J.P., Da Prato, G.: Stochastic Nagumo’s viability theorem. Stoch. Anal. Appl. 13, 1–11 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aubin, J.P., Da Prato, G.: The viability theorem for stochastic differential inclusion. Stoch. Anal. Appl. 16(1), 1–15 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  5. 5.
    Bai, X., Lin, Y.: On the existence and uniqueness of solutions to stochastic differential equations driven by \(G\)-Brownian motion with integral-Lipschitz coefficients. Acta Math. Sin. Engl. Ser. 30(3), 589–610 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buckdahn, R., Peng, S., Quincampoix, M., Rainer, C.: Existence of stochastic control under state constraints. C. R. Acad. Sci. Paris 327, 17–22 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buckdahn, R., Quincampoix, M., Rainer, C., Răşcanu, A.: Viability of moving sets for stochastic differential equation. Adv. Differ. Equ. 7(9), 1045–1072 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Buckdahn, R., Quincampoix, M., Răşcanu, A.: Viability property for backward stochastic differential equation and application to partial differential equation. Probab. Theory Relat. Fields 116(4), 485–504 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ciotir, I., Răşcanu, A.: Viability for stochastic differential equation driven by fractional Brownian motions. J. Differ. Equ. 247, 1505–1528 (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Da Prato, G., Frankowska, H.: Stochastic viability for compact sets in terms of the distance function. Dyn. Syst. Appl. 10, 177–184 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Da Prato, G., Frankowska, H.: Stochastic viability of convex sets. J. Math. Anal. Appl. 333, 151–163 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Denis, L., Hu, M., Peng, S.: Function spaces and capacity related to a sublinear expectation: application to \(G\)-Brownian motion paths. Potential Anal. 34(2), 139–161 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16, 827–852 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gao, F.: Pathwise properties and homomorphic flows for stochastic differential equations driven by \(G\)-Brownian motion. Stoch. Process. Appl. 119, 3356–3382 (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hu, M., Peng, S.: On representation theorem of \(G\)-expectations and paths of \(G\)-Brownian motion. Acta Math. Appl. Sin. Engl. Ser. 25(3), 539–546 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hu, Y., Peng, S.: On the comparison theorem for multidimensional BSDEs. Comptes Rendus Math. 343(2), 135–140 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lin, Q.: Some properties of stochastic differential equations driven by \(G\)-Brownian motion. Acta Math. Sin. Engl. Ser. 29, 923–942 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, X., Peng, S.: Stopping times and related Itô calculus with \(G\)-Brownian motion. Stoch. Process. Appl. 121, 1492–1508 (2011)CrossRefzbMATHGoogle Scholar
  19. 19.
    Li, X., Lin, X., Lin, Y.: Lyapunov-type conditions and stochastic differential equations driven by \(G\)-Brownian motion. J. Math. Anal. Appl. 439, 235–255 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Luo, P., Wang, F.: On the comparison theorem for multi-dimensional \(G\)-SDEs. Stat. Probab. Lett. 96, 38–44 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Milian, A.: A note on stochastic invariance for Itô equations. Bull. Pol. Acad. Sci. Math. 41(2), 139–150 (1993)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Nagumo, M.: Über die lage der integralkurven gewöhnlicher differentialgleichungen. Proc. Phys. Math. Soc. Jpn. 24, 551–559 (1942)zbMATHGoogle Scholar
  23. 23.
    Nie, T., Răşcanu, A.: Deterministic characterization of viability for stochastic differential equation driven by fractional Brownian motion. ESAIM Control Optim. Calc. Var. 18, 915–929 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Peng, S.: Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. Engl. Ser. 20(2), 1–24 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Peng, S.: Nonlinear expectations and nonlinear Markov chains. Chin. Ann. Math. 26B(2), 159–184 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Peng, S.: Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Ser. A Math. 52(7), 1391–1411 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty (2010). arXiv:1002.4546v1
  28. 28.
    Peng, S., Zhu, X.: Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations. Stoch. Process. Appl. 116, 370–380 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Zhongtai Securities Institute for Financial StudiesShandong UniversityJinanChina
  2. 2.Department of MathematicsETH ZurichZurichSwitzerland
  3. 3.Zhongtai Securities Institute for Financial Studies and Institute for Advanced ResearchShandong UniversityJinanChina

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