Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T15:18:54.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Adding constraints to BSDEs with jumps: an alternative tomultidimensional reflections∗

Published online by Cambridge University Press:  01 July 2014

Romuald Elie  and
Idris Kharroubi
Romuald Elie
Affiliation:
CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, Place du maréchal de Lattre de Tassigny, 75016 Paris, France. [email protected],[email protected]
Idris Kharroubi
Affiliation:
CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, Place du maréchal de Lattre de Tassigny, 75016 Paris, France. [email protected],[email protected]
Get access

Abstract

This paper is dedicated to the analysis of backward stochastic differential equations(BSDEs) with jumps, subject to an additional global constraint involving all thecomponents of the solution. We study the existence and uniqueness of a minimal solutionfor these so-called constrained BSDEs with jumps via a penalizationprocedure. This new type of BSDE offers a nice and practical unifying framework to thenotions 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]. Moreremarkably, the solution of a multidimensional Brownian reflected BSDE studied in [Y. Huand 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-dimensionalconstrained BSDE with jumps. This last result is very promising from a numerical point ofview for the resolution of high dimensional optimal switching problems and more generallyfor systems of coupled variational inequalities.

Keywords

Stochastic controlswitching problemsBSDE with jumpsreflected BSDE
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 233 - 250
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barles, G., Buckdahn, R. and Pardoux, E., Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Reports 60 (1997) 57–83. Google Scholar
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).
Bouchard, B., A stochastic target formulation for optimal switching problems in finite horizon. Stochastics 81 (2009) 171–197. Google Scholar
Bouchard, B. and Elie, R., Discrete–time approximation of decoupled forward-backward SDE with jumps. Stoch. Proc. Appl. 118 (2008) 53–75. Google Scholar
Chassagneux, J.F., Elie, R. and Kharroubi, I., A note on existence and uniqueness of multidimensional reflected BSDEs. Electronic Commun. Prob. 16 (2011) 120–128. Google Scholar
Buckdahn, R. and Hu, Y., Pricing of American contingent claims with jump stock price and constrained portfolios. Math. Oper. Res. 23 (1998) 177–203. Google Scholar
Buckdahn, R. and Hu, Y., Hedging contingent claims for a large investor in an incomplete market. Adv. Appl. Probab. 30 (1998) 239–255. Google Scholar
Buckdahn, R., Quincampoix, M. and Rascanu, A., Viability property for a backward stochastic differential equation and applications to partial differential equations. Probab. Theory Relat. Fields 116 (2000) 485–504. Google Scholar
Cvitanic, J., Karatzas, I. and Soner, M., Backward stochastic differential equations with constraints on the gain-process. Ann. Probab. 26 (1998) 1522–1551. Google Scholar
Djehiche, B., Hamadène, S. and Popier, A., The finite horizon optimal multiple switching problem. SIAM J. Control Optim. 48 (2009) 2751–2770. Google Scholar
El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M.C., Reflected solutions of Backward SDE’s, and related obstacle problems for PDEs. Ann. Prob. 25 (1997) 702–737. Google Scholar
Elie, R. and Kharroubi, I., Probabilistic representation and approximation for coupled systems of variational inequalities. Stat. Probab. Lett. 80 (2009) 1388–1396. Google Scholar
Essaky, E., Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bull. Sci. Math. 132 (2008) 690–710. Google Scholar
Hamadène, S. and Zhang, J., Switching problem and related system of reflected BSDEs. Stoch. Proc. Appl. 120 (2010) 403–426. Google Scholar
Hu, Y. and Peng, S., On comparison theorem for multi-dimensional BSDEs. C. R. Acad. Sci. Paris 343 (2006) 135–140. Google Scholar
Hu, Y. and Tang, S., Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Relat. Fields 147 (2010) 89–121. Google Scholar
Kharroubi, I., Ma, J., Pham, H. and Zhang, J., Backward SDEs with constrained jumps and Quasi–Variational Inequalities. Ann. Probab. 38 (2008) 794–840. Google Scholar
Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control. Lett. 14 (1990) 55–61. Google Scholar
Pardoux, E., Pradeilles, F. and Rao, Z., Probabilistic interpretation of a system of semi-linear parabolic partial differential equations. Ann. Inst. Henri Poincaré, Section B 33 (1997) 467–490. Google Scholar
Peng, S., Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Relat. Fields 113 (1999) 473–499. Google Scholar
Peng, S. and Xu, M., The smallest g-supermartingale and reflected BSDE with single and double obstacles. Ann. Inst. Henri Poincaré 41 (2005) 605–630. Google Scholar
S. Peng and M. Xu, Constrained BSDE and viscosity solutions of variation inequalities. Preprint. (2007).
Ramasubramanian, S., Reflected backward stochastic differential equations in an orthant. Proc. Indian Acad. Sci. 112 (2002) 347–360. Google Scholar
Royer, M. Backward stochastic differential equations with jumps and related nonlinear expectations. Stoch. Proc. Appl. 116 (2006) 1358–1376. Google Scholar
Tang, S. and Li, X., Necessary conditions for optimal control of stochastic systems with jumps. SIAM J. Control Optim. 32 (1994) 1447–1475. Google Scholar
Tang, S. and Yong, J., Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach. Stoch. Stoch. Reports 45 (1993) 145–176. Google Scholar