Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections

Romuald Elie; Idris Kharroubi

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 233-250
  • ISSN: 1292-8100

Abstract

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This paper is dedicated to the analysis of backward stochastic differential equations (BSDEs) with jumps, subject to an additional global constraint involving all the components of the solution. We study the existence and uniqueness of a minimal solution for these so-called constrained BSDEs with jumps via a penalization procedure. This new type of BSDE offers a nice and practical unifying framework to the notions of constrained BSDEs presented in [S. Peng and M. Xu, Preprint. (2007)] and BSDEs with constrained jumps introduced in [I. Kharroubi, J. Ma, H. Pham and J. Zhang, Ann. Probab. 38 (2008) 794–840]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [Y. Hu and S. Tang, Probab. Theory Relat. Fields 147 (2010) 89–121] and [S. Hamadène and J. Zhang, Stoch. Proc. Appl. 120 (2010) 403–426] can also be represented via a well chosen one-dimensional constrained BSDE with jumps. This last result is very promising from a numerical point of view for the resolution of high dimensional optimal switching problems and more generally for systems of coupled variational inequalities.

How to cite

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Elie, Romuald, and Kharroubi, Idris. "Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections." ESAIM: Probability and Statistics 18 (2014): 233-250. <http://eudml.org/doc/273641>.

@article{Elie2014,
abstract = {This paper is dedicated to the analysis of backward stochastic differential equations (BSDEs) with jumps, subject to an additional global constraint involving all the components of the solution. We study the existence and uniqueness of a minimal solution for these so-called constrained BSDEs with jumps via a penalization procedure. This new type of BSDE offers a nice and practical unifying framework to the notions of constrained BSDEs presented in [S. Peng and M. Xu, Preprint. (2007)] and BSDEs with constrained jumps introduced in [I. Kharroubi, J. Ma, H. Pham and J. Zhang, Ann. Probab. 38 (2008) 794–840]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [Y. Hu and S. Tang, Probab. Theory Relat. Fields 147 (2010) 89–121] and [S. Hamadène and J. Zhang, Stoch. Proc. Appl. 120 (2010) 403–426] can also be represented via a well chosen one-dimensional constrained BSDE with jumps. This last result is very promising from a numerical point of view for the resolution of high dimensional optimal switching problems and more generally for systems of coupled variational inequalities.},
author = {Elie, Romuald, Kharroubi, Idris},
journal = {ESAIM: Probability and Statistics},
keywords = {stochastic control; switching problems; BSDE with jumps; reflected BSDE; backward stochastic differential equations (BSDEs) with jumps},
language = {eng},
pages = {233-250},
publisher = {EDP-Sciences},
title = {Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections},
url = {http://eudml.org/doc/273641},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Elie, Romuald
AU - Kharroubi, Idris
TI - Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 233
EP - 250
AB - This paper is dedicated to the analysis of backward stochastic differential equations (BSDEs) with jumps, subject to an additional global constraint involving all the components of the solution. We study the existence and uniqueness of a minimal solution for these so-called constrained BSDEs with jumps via a penalization procedure. This new type of BSDE offers a nice and practical unifying framework to the notions of constrained BSDEs presented in [S. Peng and M. Xu, Preprint. (2007)] and BSDEs with constrained jumps introduced in [I. Kharroubi, J. Ma, H. Pham and J. Zhang, Ann. Probab. 38 (2008) 794–840]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [Y. Hu and S. Tang, Probab. Theory Relat. Fields 147 (2010) 89–121] and [S. Hamadène and J. Zhang, Stoch. Proc. Appl. 120 (2010) 403–426] can also be represented via a well chosen one-dimensional constrained BSDE with jumps. This last result is very promising from a numerical point of view for the resolution of high dimensional optimal switching problems and more generally for systems of coupled variational inequalities.
LA - eng
KW - stochastic control; switching problems; BSDE with jumps; reflected BSDE; backward stochastic differential equations (BSDEs) with jumps
UR - http://eudml.org/doc/273641
ER -

References

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