Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections
ESAIM: Probability and Statistics, Volume 18 (2014), p. 233-250
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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.

DOI : https://doi.org/10.1051/ps/2013036
Classification:  93E20,  60H30,  60J75
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     pages = {233-250},
     doi = {10.1051/ps/2013036},
     mrnumber = {3230876},
     language = {en},
     url = {http://www.numdam.org/item/PS_2014__18__233_0}
}
Elie, Romuald; Kharroubi, Idris. Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections. ESAIM: Probability and Statistics, Volume 18 (2014) pp. 233-250. doi : 10.1051/ps/2013036. http://www.numdam.org/item/PS_2014__18__233_0/

[1] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Reports 60 (1997) 57-83. | MR 1436432 | Zbl 0878.60036

[2] M. Bernhart, H. Pham, P. Tankov and X. Warin, Swing Options Valuation: a BSDE with Constrained Jumps Approach. Numerical methods in finance. Edited by R. Carmona et al. Springer (2012). | Zbl 1247.91179

[3] B. Bouchard, A stochastic target formulation for optimal switching problems in finite horizon. Stochastics 81 (2009) 171-197. | MR 2571686 | Zbl 1175.60037

[4] B. Bouchard and R. Elie, Discrete-time approximation of decoupled forward-backward SDE with jumps. Stoch. Proc. Appl. 118 (2008) 53-75. | MR 2376252 | Zbl 1136.60048

[5] J.F. Chassagneux, R. Elie and I. Kharroubi, A note on existence and uniqueness of multidimensional reflected BSDEs. Electronic Commun. Prob. 16 (2011) 120-128. | MR 2775350 | Zbl 1232.93095

[6] R. Buckdahn and Y. Hu, Pricing of American contingent claims with jump stock price and constrained portfolios. Math. Oper. Res. 23 (1998) 177-203. | MR 1606470 | Zbl 0985.91026

[7] Buckdahn R. and Y. Hu, Hedging contingent claims for a large investor in an incomplete market. Adv. Appl. Probab. 30 (1998) 239-255. | MR 1618845 | Zbl 0904.90009

[8] R. Buckdahn, M. Quincampoix and A. Rascanu, Viability property for a backward stochastic differential equation and applications to partial differential equations. Probab. Theory Relat. Fields 116 (2000) 485-504. | MR 1757597 | Zbl 0969.60061

[9] J. Cvitanic, I. Karatzas and M. Soner, Backward stochastic differential equations with constraints on the gain-process. Ann. Probab. 26 (1998) 1522-1551. | MR 1675035 | Zbl 0935.60039

[10] B. Djehiche, S. Hamadène and A. Popier, The finite horizon optimal multiple switching problem. SIAM J. Control Optim. 48 (2009) 2751-2770. | MR 2558319 | Zbl 1196.60069

[11] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M.C. Quenez, Reflected solutions of Backward SDE's, and related obstacle problems for PDEs. Ann. Prob. 25 (1997) 702-737. | MR 1434123 | Zbl 0899.60047

[12] R. Elie and I. Kharroubi, Probabilistic representation and approximation for coupled systems of variational inequalities. Stat. Probab. Lett. 80 (2009) 1388-1396. | MR 2669778 | Zbl 1194.93218

[13] E. Essaky, Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bull. Sci. Math. 132 (2008) 690-710. | MR 2474488 | Zbl 1157.60057

[14] S. Hamadène and J. Zhang, Switching problem and related system of reflected BSDEs. Stoch. Proc. Appl. 120 (2010) 403-426. | MR 2594364 | Zbl 1191.60056

[15] Y. Hu and S. Peng, On comparison theorem for multi-dimensional BSDEs. C. R. Acad. Sci. Paris 343 (2006) 135-140. | MR 2243308 | Zbl 1098.60052

[16] Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Relat. Fields 147 (2010) 89-121. | MR 2594348 | Zbl 1188.60029

[17] I. Kharroubi, J. Ma, H. Pham and J. Zhang, Backward SDEs with constrained jumps and Quasi-Variational Inequalities. Ann. Probab. 38 (2008) 794-840. | MR 2642892 | Zbl 1205.60114

[18] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control. Lett. 14 (1990) 55-61. | MR 1037747 | Zbl 0692.93064

[19] E. Pardoux, F. Pradeilles and Z. Rao, Probabilistic interpretation of a system of semi-linear parabolic partial differential equations. Ann. Inst. Henri Poincaré, Section B 33 (1997) 467-490. | Numdam | MR 1465798 | Zbl 0891.60054

[20] S. Peng, Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probab. Theory Relat. Fields 113 (1999) 473-499. | MR 1717527 | Zbl 0953.60059

[21] S. Peng and M. Xu, The smallest g-supermartingale and reflected BSDE with single and double obstacles. Ann. Inst. Henri Poincaré 41 (2005) 605-630. | Numdam | MR 2139035 | Zbl 1071.60049

[22] S. Peng and M. Xu, Constrained BSDE and viscosity solutions of variation inequalities. Preprint. (2007).

[23] S. Ramasubramanian, Reflected backward stochastic differential equations in an orthant. Proc. Indian Acad. Sci. 112 (2002) 347-360. | MR 1908376 | Zbl 1007.60061

[24] M. Royer Backward stochastic differential equations with jumps and related nonlinear expectations. Stoch. Proc. Appl. 116 (2006) 1358-1376. | MR 2260739 | Zbl 1110.60062

[25] S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with jumps. SIAM J. Control Optim. 32 (1994) 1447-1475. | MR 1288257 | Zbl 0922.49021

[26] S. Tang and J. Yong, Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach. Stoch. Stoch. Reports 45 (1993) 145-176. | MR 1306930 | Zbl 0795.93103