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.
- V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Ed. Academiei Române and Noordhoff International Publishing, 1976.
- C. Bender, Explicit solutions of a class of linear fractional BSDEs, Systems Control Lett. 54 (2005) 7, 671-680.
- 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.
- 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.
- D. Borkowski, K. Jańczak-Borkowska, BSDE driven by a multidimensional fractional Brownian motion, submitted.
- H. Brézis, Opérateurs maximaux monotones et semigroupes de contractions dans les spaces de Hilbert, North-Holland Publ. Co., 1973.
- 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.
- L. Decreusefond, A.S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1998), 177-214.
- T.E. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motions. I. Theory, SIAM J. Control Optim. 38 (2000), 582-612.
- Y. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Mem. Amer. Math. Soc. 175 (2005) 825.
- Y. Hu, B. Øksendal, Fractional white noise calculus and application to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 1-32.
- Y. Hu, S. Peng, Backward stochastic differential equation driven by fractional Brownian motion, Siam J. Control Optim. 48 (2009) 3, 1675-1700.
- K. Jańczak-Borkowska, Generalized BSDEs driven by fractional Brownian motion, Statist. Probab. Lett. 83 (2013) 3, 805-811.
- S.J. Lin, Stochastic analysis of fractional Brownian motions, Stochastics Stochastics Rep. 55 (1995), 121-140.
- L. Maticiuc, T. Nie, Fractional backward stochastic differential equations and fractional backward variational inequalities, J. Theoret. Probab. 28 (2015) 1, 337-395.
- L. Maticiuc, A. Răşcanu, A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl. 120 (2010) 6, 777-800.
- J. Miao, X. Yang, Solutions to BSDEs driven by multidimensional fractional Brownian motions, Math. Probl. Eng. 2015 (2015), Article ID 481842.
- D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Springer, Berlin, 2010.
- É. Pardoux, S. Peng, Adapted solutions of a backward stochastic differential equation, Systems Control Lett. 14 (1990), 55-61.
- É. Pardoux, A. Răşcanu, Stochastic differential equations, Backward SDEs, Partial differential equations, Springer International Publishing, 2014.
- 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.
