Multivalued backward stochastic differential equations with time delayed generators

Bakarime Diomande; Lucian Maticiuc

Open Mathematics (2014)

  • Volume: 12, Issue: 11, page 1624-1637
  • ISSN: 2391-5455

Abstract

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Our aim is to study the following new type of multivalued backward stochastic differential equation: - d Y t + ∂ φ Y t d t ∋ F t , Y t , Z t , Y t , Z t d t + Z t d W t , 0 ⩽ t ⩽ T , Y T = ξ , where ∂φ is the subdifferential of a convex function and (Y t, Z t):= (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.

How to cite

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Bakarime Diomande, and Lucian Maticiuc. "Multivalued backward stochastic differential equations with time delayed generators." Open Mathematics 12.11 (2014): 1624-1637. <http://eudml.org/doc/269286>.

@article{BakarimeDiomande2014,
abstract = {Our aim is to study the following new type of multivalued backward stochastic differential equation: \[\left\lbrace \{\begin\{array\}\{c\}- dY\left( t \right) + \partial \phi \left( \{Y\left( t \right)\} \right)dt \ni F\left( \{t,Y\left( t \right),Z\left( t \right),Y\_t ,Z\_t \} \right)dt + Z\left( t \right)dW\left( t \right), 0 \leqslant t \leqslant T, \hfill \\ Y\left( T \right) = \xi , \hfill \\ \end\{array\}\} \right.\] where ∂φ is the subdifferential of a convex function and (Y t, Z t):= (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.},
author = {Bakarime Diomande, Lucian Maticiuc},
journal = {Open Mathematics},
keywords = {Backward stochastic differential equations; Time-delayed generators; Subdifferential operator; backward stochastic differential equations; time-delayed generators; subdifferential operator},
language = {eng},
number = {11},
pages = {1624-1637},
title = {Multivalued backward stochastic differential equations with time delayed generators},
url = {http://eudml.org/doc/269286},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Bakarime Diomande
AU - Lucian Maticiuc
TI - Multivalued backward stochastic differential equations with time delayed generators
JO - Open Mathematics
PY - 2014
VL - 12
IS - 11
SP - 1624
EP - 1637
AB - Our aim is to study the following new type of multivalued backward stochastic differential equation: \[\left\lbrace {\begin{array}{c}- dY\left( t \right) + \partial \phi \left( {Y\left( t \right)} \right)dt \ni F\left( {t,Y\left( t \right),Z\left( t \right),Y_t ,Z_t } \right)dt + Z\left( t \right)dW\left( t \right), 0 \leqslant t \leqslant T, \hfill \\ Y\left( T \right) = \xi , \hfill \\ \end{array}} \right.\] where ∂φ is the subdifferential of a convex function and (Y t, Z t):= (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.
LA - eng
KW - Backward stochastic differential equations; Time-delayed generators; Subdifferential operator; backward stochastic differential equations; time-delayed generators; subdifferential operator
UR - http://eudml.org/doc/269286
ER -

References

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  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. http://dx.doi.org/10.1214/09-AAP663 Zbl1202.34144
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  7. [7] Lucian Maticiuc, Aurel Răşcanu, A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl. 120 (2010), no. 6, 777–800. http://dx.doi.org/10.1016/j.spa.2010.02.002 Zbl1195.35192
  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). Zbl1332.60085
  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. http://dx.doi.org/10.2478/s11533-011-0131-y Zbl1257.65006
  10. [10] Etienne Pardoux, Shige Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55–61. http://dx.doi.org/10.1016/0167-6911(90)90082-6 
  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). 
  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. http://dx.doi.org/10.1016/S0304-4149(98)00030-1 Zbl0932.60070
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