Opuscula Math. 38, no. 3 (2018), 307-326
https://doi.org/10.7494/OpMath.2018.38.3.307

 
Opuscula Mathematica

Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion

Dariusz Borkowski
Katarzyna Jańczak-Borkowska

Abstract. We study the existence and uniqueness of the backward stochastic variational inequalities driven by \(m\)-dimensional fractional Brownian motion with Hurst parameters \(H_k\) (\(k=1,\ldots m\)) greater than \(1/2\). The stochastic integral used throughout the paper is the divergence type integral.

Keywords: backward stochastic differential equation, fractional Brownian motion, backward stochastic variational inequalities, subdifferential operator.

Mathematics Subject Classification: 60H05, 60H07, 60H22.

Full text (pdf)

  1. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Ed. Academiei Române and Noordhoff International Publishing, 1976.
  2. C. Bender, Explicit solutions of a class of linear fractional BSDEs, Systems Control Lett. 54 (2005) 7, 671-680.
  3. F. Biagini, Y. Hu, B. Øksendal, A. Sulem, A stochastic maximum principle for processes driven by fractional Brownian motion, Stochastic Process. Appl. 100 (2002) 1, 233-253.
  4. D. Borkowski, K. Jańczak-Borkowska, Generalized backward stochastic variational inequalities driven by a fractional Brownian motion, Braz. J. Probab. Stat. 30 (2016) 3, 502-519.
  5. D. Borkowski, K. Jańczak-Borkowska, BSDE driven by a multidimensional fractional Brownian motion, submitted.
  6. H. Brézis, Opérateurs maximaux monotones et semigroupes de contractions dans les spaces de Hilbert, North-Holland Publ. Co., 1973.
  7. W. Dai, C.C. Heyde, Itô formula with respect to fractional Brownian motion and its application, J. Appl. Math. Stoch. Anal. 9 (1990), 439-448.
  8. L. Decreusefond, A.S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1998), 177-214.
  9. T.E. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motions. I. Theory, SIAM J. Control Optim. 38 (2000), 582-612.
  10. Y. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Mem. Amer. Math. Soc. 175 (2005) 825.
  11. Y. Hu, B. Øksendal, Fractional white noise calculus and application to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 1-32.
  12. Y. Hu, S. Peng, Backward stochastic differential equation driven by fractional Brownian motion, Siam J. Control Optim. 48 (2009) 3, 1675-1700.
  13. K. Jańczak-Borkowska, Generalized BSDEs driven by fractional Brownian motion, Statist. Probab. Lett. 83 (2013) 3, 805-811.
  14. S.J. Lin, Stochastic analysis of fractional Brownian motions, Stochastics Stochastics Rep. 55 (1995), 121-140.
  15. L. Maticiuc, T. Nie, Fractional backward stochastic differential equations and fractional backward variational inequalities, J. Theoret. Probab. 28 (2015) 1, 337-395.
  16. L. Maticiuc, A. Răşcanu, A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl. 120 (2010) 6, 777-800.
  17. J. Miao, X. Yang, Solutions to BSDEs driven by multidimensional fractional Brownian motions, Math. Probl. Eng. 2015 (2015), Article ID 481842.
  18. D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Springer, Berlin, 2010.
  19. É. Pardoux, S. Peng, Adapted solutions of a backward stochastic differential equation, Systems Control Lett. 14 (1990), 55-61.
  20. É. Pardoux, A. Răşcanu, Stochastic differential equations, Backward SDEs, Partial differential equations, Springer International Publishing, 2014.
  21. L.C. Young, An inequality of the Hölder type connected with Stieltjes integration, Acta Math. 67 (1936), 251-282.
  • Dariusz Borkowski
  • Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toruń, Poland
  • Katarzyna Jańczak-Borkowska
  • University of Science and Technology, Institute of Mathematics and Physics, al. prof. S. Kaliskiego 7, 85-796 Bydgoszcz, Poland
  • Communicated by Tomasz Zastawniak.
  • Received: 2017-03-07.
  • Revised: 2017-10-22.
  • Accepted: 2017-11-17.
  • Published online: 2018-03-19.
Opuscula Mathematica - cover

Cite this article as:
Dariusz Borkowski, Katarzyna Jańczak-Borkowska, Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion, Opuscula Math. 38, no. 3 (2018), 307-326, https://doi.org/10.7494/OpMath.2018.38.3.307

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.