Central European Journal of Mathematics

, Volume 12, Issue 11, pp 1624–1637 | Cite as

Multivalued backward stochastic differential equations with time delayed generators

Research Article


Our aim is to study the following new type of multivalued backward stochastic differential equation:
$$\left\{ \begin{gathered} - dY\left( t \right) + \partial \phi \left( {Y\left( t \right)} \right)dt \ni F\left( {t,Y\left( t \right),Z\left( t \right),Y_t ,Z_t } \right)dt + Z\left( t \right)dW\left( t \right), 0 \leqslant t \leqslant T, \hfill \\ Y\left( T \right) = \xi , \hfill \\ \end{gathered} \right.$$
where ∂φ is the subdifferential of a convex function and (Yt, Zt):= (Y(t + θ), Z(t + θ))θ∈[−T,0] represent the past values of the solution over the interval [0, t]. Our results are based on the existence theorem from Delong & Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.


Backward stochastic differential equations Time-delayed generators Subdifferential operator 


60H10 47J20 49J40 


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  1. [1]
    Haïm Brézis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.Google Scholar
  2. [2]
    Łukasz Delong, Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management, preprint, 2011 (http://arxiv.org/abs/1005.4417).Google Scholar
  3. [3]
    Łukasz Delong, Peter Imkeller, Backward stochastic differential equations with time delayed generators — results and counterexamples, The Annals of Applied Probability 20 (2010), no. 4, 1512–1536.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    Łukasz Delong, Peter Imkeller, On Malliavin’s differentiability of BSDE with time delayed generators driven by Brownian motions and Poisson random measures, Stochastic Process. Appl. 120 (2010), no. 9, 1748–1775.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    Gonçalo dos Reis, Anthony Réveillac, Jianing Zhang, FBSDEs with time delayed generators: Lp-solutions, differentiability, representation formulas and path regularity, Stochastic Process. Appl. 121 (2011), no. 9, 2114–2150.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    Nicole El Karoui, Christophe Kapoudjian, Etienne Pardoux, Shige Peng, Marie-Claire Quenez, Reflected solutions of backward SDE’s and related obstacle problems for PDE’s, Ann.Probab. 25 (1997), no. 2, 702–737.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    Lucian Maticiuc, Aurel Răşcanu, A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl. 120 (2010), no. 6, 777–800.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Lucian Maticiuc, Aurel Răşcanu, Backward Stochastic Variational Inequalities on Random Interval, accepted for publication in Bernoulli, 2014 (http://arxiv.org/abs/1112.5792).Google Scholar
  9. [9]
    Lucian Maticiuc, Eduard Rotenstein, Numerical schemes for multivalued backward stochastic differential systems, Central European Journal of Mathematics 10 (2012), no. 2, 693–702.CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    Etienne Pardoux, Shige Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55–61.CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    Etienne Pardoux, Shige Peng, Backward SDE’s and quasilinear parabolic PDE’s, Stochastic PDE and Their Applications (B.L. Rozovskii, R.B. Sowers eds.), 200–217, LNCIS 176, Springer (1992).Google Scholar
  12. [12]
    Etienne Pardoux, Aurel Răşcanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl. 76 (1998), no. 2, 191–215.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    Aurel Răşcanu, Eduard Rotenstein, The Fitzpatrick function-a bridge between convex analysis and multivalued stochastic differential equations, Journal of Convex Analysis 18 (2011), no. 1, 105–138.MATHMathSciNetGoogle Scholar
  14. [14]
    Qing Zhou, Yong Ren, Reflected backward stochastic differential equations with time delayed generators, Statistics and Probability Letters 82 (2012), no. 5, 979–990.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Versita Warsaw and Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Faculty of Mathematics“Alexandru Ioan Cuza” UniversityIaşiRomania
  2. 2.Department of Mathematics“Gheorghe Asachi” Technical UniversityIaşiRomania

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