Central European Journal of Mathematics

, Volume 10, Issue 2, pp 693–702 | Cite as

Numerical schemes for multivalued backward stochastic differential systems

Research Article

Abstract

We define approximation schemes for generalized backward stochastic differential systems, considered in the Markovian framework. More precisely, we propose a mixed approximation scheme for the following backward stochastic variational inequality:
$$dY_t + F(t,X_t ,Y_t ,Z_t )dt \in \partial \phi (Y_t )dt + Z_t dW_t ,$$
where ∂φ is the subdifferential operator of a convex lower semicontinuous function φ and (Xt)t∈[0;T] is the unique solution of a forward stochastic differential equation. We use an Euler type scheme for the system of decoupled forward-backward variational inequality in conjunction with Yosida approximation techniques.

Keywords

Euler scheme Yosida approximation Error estimate Multivalued backward SDEs Reflected SDEs 

MSC

65C99 60H30 47H15 

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References

  1. [1]
    Asiminoaei I., Rascanu A., Approximation and simulation of stochastic variational inequalities — splitting up method, Numer. Funct. Anal. Optim., 1997, 18(3–4), 251–282MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Bouchard B., Menozzi S., Strong approximations of BSDEs in a domain, Bernoulli, 2009, 15(4), 1117–1147MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Bouchard B., Touzi N., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 2004, 111(2), 175–206MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Chitashvili R.J., Lazrieva N.L., Strong solutions of stochastic differential equations with boundary conditions, Stochastics, 1981, 5(4), 225–309MathSciNetCrossRefGoogle Scholar
  5. [5]
    Constantini C., Pacchiarotti B., Sartoretto F., Numerical approximation for functionals of reflecting diffusion processes, SIAM J. Appl. Math., 1998, 58(1), 73–102MathSciNetCrossRefGoogle Scholar
  6. [6]
    Ding D., Zhang Y.Y., A splitting-step algorithm for reflected stochastic differential equations in ℝ+1, Comput. Math. Appl., 2008, 55(11), 2413–2425MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Karatzas I., Shreve S.E., Brownian Motion and Stochastic Calculus, Grad. Texts in Math., 113, Springer, New York, 1988Google Scholar
  8. [8]
    Kloeden P.E., Platen E., Numerical Solution of Stochastic Differential Equations, Appl. Math. (N. Y.), Springer, Berlin, 1992Google Scholar
  9. [9]
    Lépingle D., Euler scheme for reflected stochastic differential equations, Math. Comput. Simulation, 1995, 38(1–3), 119–126MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Lions P.-L., Sznitman A.-S., Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 1984, 37(4), 511–537MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Maticiuc L., Răşcanu A., Backward stochastic generalized variational inequality, In: Applied Analysis and Differential Equations, Iaşi, September 4–9, 2006, World Scientific, Hackensack, 2007, 217–226CrossRefGoogle Scholar
  12. [12]
    Maticiuc L., Răşcanu A., A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl., 2010, 120(6), 777–800MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Menaldi J.-L., Stochastic variational inequality for reflected diffusion, Indiana Univ. Math. J., 1983, 32(5), 733–744MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Pardoux É., Peng S.G., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 1990, 14(1), 55–61MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Pardoux É., Peng S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, In: Stochastic Partial Differential Equations and their Applications, Charlotte, June 6–8, 1991, Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992, 200–217CrossRefGoogle Scholar
  16. [16]
    Pardoux E., Răşcanu A., Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl., 1998, 76(2), 191–215MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Pardoux E., Răşcanu A., Backward stochastic variational inequalities, Stochastics Stochastics Rep., 1999, 67(3–4), 159–167MathSciNetMATHGoogle Scholar
  18. [18]
    Rascanu A., Deterministic and stochastic differential equations in Hilbert spaces involving multivalued maximal monotone operators, Panamer. Math. J., 1996, 6(3), 83–119MathSciNetMATHGoogle Scholar
  19. [19]
    Răşcanu A., Rotenstein E., The Fitzpatrick function - a bridge between convex analysis and multivalued stochastic differential equations, J. Convex Anal., 2011, 18(1), 105–138MathSciNetMATHGoogle Scholar
  20. [20]
    Saisho Y., Stochastic differential equations for multidimensional domain with reflecting boundary, Probab. Theory Related Fields, 1987, 74(3), 455–477MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Skorokhod A.V., Stochastic equations for diffusion processes in a bounded region. I&II, Theory Probab. Appl., 1961, 6(3), 264–274; 7(1), 3–23CrossRefGoogle Scholar
  22. [22]
    SŁominski L., On approximation of solutions of multidimensional SDEs with reflecting boundary conditions, Stochastic Process. Appl., 1994, 50(2), 179–219Google Scholar
  23. [23]
    Zhang J., Some Fine Properties of Backward Stochastic Differential Equations, PhD thesis, Purdue University, 2001Google Scholar

Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2011

Authors and Affiliations

  1. 1.Faculty of MathematicsAlexandru Ioan Cuza UniversityIaşiRomania

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