# Numerical schemes for multivalued backward stochastic differential systems

Lucian Maticiuc; Eduard Rotenstein

Open Mathematics (2012)

- Volume: 10, Issue: 2, page 693-702
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topLucian Maticiuc, and Eduard Rotenstein. "Numerical schemes for multivalued backward stochastic differential systems." Open Mathematics 10.2 (2012): 693-702. <http://eudml.org/doc/269491>.

@article{LucianMaticiuc2012,

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 (X t)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.},

author = {Lucian Maticiuc, Eduard Rotenstein},

journal = {Open Mathematics},

keywords = {Euler scheme; Yosida approximation; Error estimate; Multivalued backward SDEs; Reflected SDEs; error estimate; multivalued backward SDEs; reflected SDEs; Brownian motion; stochastic differential equations (SDEs)},

language = {eng},

number = {2},

pages = {693-702},

title = {Numerical schemes for multivalued backward stochastic differential systems},

url = {http://eudml.org/doc/269491},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Lucian Maticiuc

AU - Eduard Rotenstein

TI - Numerical schemes for multivalued backward stochastic differential systems

JO - Open Mathematics

PY - 2012

VL - 10

IS - 2

SP - 693

EP - 702

AB - 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 (X t)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.

LA - eng

KW - Euler scheme; Yosida approximation; Error estimate; Multivalued backward SDEs; Reflected SDEs; error estimate; multivalued backward SDEs; reflected SDEs; Brownian motion; stochastic differential equations (SDEs)

UR - http://eudml.org/doc/269491

ER -

## References

top- [1] Asiminoaei I., Rascanu A., Approximation and simulation of stochastic variational inequalities - splitting up method, Numer. Funct. Anal. Optim., 1997, 18(3–4), 251–282 http://dx.doi.org/10.1080/01630569708816759 Zbl0883.60057
- [2] Bouchard B., Menozzi S., Strong approximations of BSDEs in a domain, Bernoulli, 2009, 15(4), 1117–1147 http://dx.doi.org/10.3150/08-BEJ181 Zbl1204.60048
- [3] Bouchard B., Touzi N., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 2004, 111(2), 175–206 http://dx.doi.org/10.1016/j.spa.2004.01.001 Zbl1071.60059
- [4] Chitashvili R.J., Lazrieva N.L., Strong solutions of stochastic differential equations with boundary conditions, Stochastics, 1981, 5(4), 225–309 http://dx.doi.org/10.1080/17442508108833184 Zbl0479.60062
- [5] Constantini C., Pacchiarotti B., Sartoretto F., Numerical approximation for functionals of reflecting diffusion processes, SIAM J. Appl. Math., 1998, 58(1), 73–102 http://dx.doi.org/10.1137/S0036139995291040 Zbl0913.60031
- [6] Ding D., Zhang Y.Y., A splitting-step algorithm for reflected stochastic differential equations in ℝ +1, Comput. Math. Appl., 2008, 55(11), 2413–2425 http://dx.doi.org/10.1016/j.camwa.2007.08.043 Zbl1142.65307
- [7] Karatzas I., Shreve S.E., Brownian Motion and Stochastic Calculus, Grad. Texts in Math., 113, Springer, New York, 1988 Zbl0638.60065
- [8] Kloeden P.E., Platen E., Numerical Solution of Stochastic Differential Equations, Appl. Math. (N. Y.), Springer, Berlin, 1992 Zbl0752.60043
- [9] Lépingle D., Euler scheme for reflected stochastic differential equations, Math. Comput. Simulation, 1995, 38(1–3), 119–126 http://dx.doi.org/10.1016/0378-4754(93)E0074-F Zbl0824.60062
- [10] Lions P.-L., Sznitman A.-S., Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 1984, 37(4), 511–537 http://dx.doi.org/10.1002/cpa.3160370408 Zbl0598.60060
- [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, 217–226 http://dx.doi.org/10.1142/9789812708229_0018 Zbl1168.60023
- [12] Maticiuc L., Răşcanu A., A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl., 2010, 120(6), 777–800 http://dx.doi.org/10.1016/j.spa.2010.02.002 Zbl1195.35192
- [13] Menaldi J.-L., Stochastic variational inequality for reflected diffusion, Indiana Univ. Math. J., 1983, 32(5), 733–744 http://dx.doi.org/10.1512/iumj.1983.32.32048 Zbl0492.60057
- [14] Pardoux É., Peng S.G., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 1990, 14(1), 55–61 http://dx.doi.org/10.1016/0167-6911(90)90082-6 Zbl0692.93064
- [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, 200–217 http://dx.doi.org/10.1007/BFb0007334
- [16] Pardoux E., Răşcanu A., Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl., 1998, 76(2), 191–215 http://dx.doi.org/10.1016/S0304-4149(98)00030-1 Zbl0932.60070
- [17] Pardoux E., Răşcanu A., Backward stochastic variational inequalities, Stochastics Stochastics Rep., 1999, 67(3–4), 159–167 Zbl0948.60049
- [18] Rascanu A., Deterministic and stochastic differential equations in Hilbert spaces involving multivalued maximal monotone operators, Panamer. Math. J., 1996, 6(3), 83–119 Zbl0859.60060
- [19] Răşcanu A., Rotenstein E., The Fitzpatrick function - a bridge between convex analysis and multivalued stochastic differential equations, J. Convex Anal., 2011, 18(1), 105–138 Zbl1210.60070
- [20] Saisho Y., Stochastic differential equations for multidimensional domain with reflecting boundary, Probab. Theory Related Fields, 1987, 74(3), 455–477 http://dx.doi.org/10.1007/BF00699100
- [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), 3–23 http://dx.doi.org/10.1137/1106035
- [22] SŁominski L., On approximation of solutions of multidimensional SDEs with reflecting boundary conditions, Stochastic Process. Appl., 1994, 50(2), 179–219 Zbl0799.60055
- [23] Zhang J., Some Fine Properties of Backward Stochastic Differential Equations, PhD thesis, Purdue University, 2001

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.