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.
Mots clés : stochastic control, switching problems, BSDE with jumps, reflected BSDE
@article{PS_2014__18__233_0,
author = {Elie, Romuald and Kharroubi, Idris},
title = {Adding constraints to {BSDEs} with jumps: an alternative to multidimensional reflections},
journal = {ESAIM: Probability and Statistics},
pages = {233--250},
publisher = {EDP-Sciences},
volume = {18},
year = {2014},
doi = {10.1051/ps/2013036},
mrnumber = {3230876},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2013036/}
}
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 SP - 233 EP - 250 VL - 18 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2013036/ DO - 10.1051/ps/2013036 LA - en ID - PS_2014__18__233_0 ER -
%0 Journal Article %A Elie, Romuald %A Kharroubi, Idris %T Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections %J ESAIM: Probability and Statistics %D 2014 %P 233-250 %V 18 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2013036/ %R 10.1051/ps/2013036 %G en %F PS_2014__18__233_0
Elie, Romuald; Kharroubi, Idris. Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 233-250. doi : 10.1051/ps/2013036. http://archive.numdam.org/articles/10.1051/ps/2013036/
[1] , and , Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Reports 60 (1997) 57-83. | MR | Zbl
[2] , , and , Swing Options Valuation: a BSDE with Constrained Jumps Approach. Numerical methods in finance. Edited by R. Carmona et al. Springer (2012). | Zbl
[3] , A stochastic target formulation for optimal switching problems in finite horizon. Stochastics 81 (2009) 171-197. | MR | Zbl
[4] and , Discrete-time approximation of decoupled forward-backward SDE with jumps. Stoch. Proc. Appl. 118 (2008) 53-75. | MR | Zbl
[5] , and , A note on existence and uniqueness of multidimensional reflected BSDEs. Electronic Commun. Prob. 16 (2011) 120-128. | MR | Zbl
[6] and , Pricing of American contingent claims with jump stock price and constrained portfolios. Math. Oper. Res. 23 (1998) 177-203. | MR | Zbl
[7] and , Hedging contingent claims for a large investor in an incomplete market. Adv. Appl. Probab. 30 (1998) 239-255. | MR | Zbl
[8] , and , Viability property for a backward stochastic differential equation and applications to partial differential equations. Probab. Theory Relat. Fields 116 (2000) 485-504. | MR | Zbl
[9] , and , Backward stochastic differential equations with constraints on the gain-process. Ann. Probab. 26 (1998) 1522-1551. | MR | Zbl
[10] , and , The finite horizon optimal multiple switching problem. SIAM J. Control Optim. 48 (2009) 2751-2770. | MR | Zbl
[11] , , , and , Reflected solutions of Backward SDE's, and related obstacle problems for PDEs. Ann. Prob. 25 (1997) 702-737. | MR | Zbl
[12] and , Probabilistic representation and approximation for coupled systems of variational inequalities. Stat. Probab. Lett. 80 (2009) 1388-1396. | MR | Zbl
[13] , Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bull. Sci. Math. 132 (2008) 690-710. | MR | Zbl
[14] and , Switching problem and related system of reflected BSDEs. Stoch. Proc. Appl. 120 (2010) 403-426. | MR | Zbl
[15] and , On comparison theorem for multi-dimensional BSDEs. C. R. Acad. Sci. Paris 343 (2006) 135-140. | MR | Zbl
[16] and , Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Relat. Fields 147 (2010) 89-121. | MR | Zbl
[17] , , and , Backward SDEs with constrained jumps and Quasi-Variational Inequalities. Ann. Probab. 38 (2008) 794-840. | MR | Zbl
[18] and , Adapted solution of a backward stochastic differential equation, Systems Control. Lett. 14 (1990) 55-61. | MR | Zbl
[19] , and , Probabilistic interpretation of a system of semi-linear parabolic partial differential equations. Ann. Inst. Henri Poincaré, Section B 33 (1997) 467-490. | Numdam | MR | Zbl
[20] , Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probab. Theory Relat. Fields 113 (1999) 473-499. | MR | Zbl
[21] and , The smallest g-supermartingale and reflected BSDE with single and double obstacles. Ann. Inst. Henri Poincaré 41 (2005) 605-630. | Numdam | MR | Zbl
[22] and , Constrained BSDE and viscosity solutions of variation inequalities. Preprint. (2007).
[23] , Reflected backward stochastic differential equations in an orthant. Proc. Indian Acad. Sci. 112 (2002) 347-360. | MR | Zbl
[24] Backward stochastic differential equations with jumps and related nonlinear expectations. Stoch. Proc. Appl. 116 (2006) 1358-1376. | MR | Zbl
[25] and , Necessary conditions for optimal control of stochastic systems with jumps. SIAM J. Control Optim. 32 (1994) 1447-1475. | MR | Zbl
[26] and , Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach. Stoch. Stoch. Reports 45 (1993) 145-176. | MR | Zbl
Cité par Sources :
