# American Institute of Mathematical Sciences

June  2013, 3(2): 233-244. doi: 10.3934/mcrf.2013.3.233

## Constrained BSDEs, viscosity solutions of variational inequalities and their applications

 1 School of Mathematics and System Science, Shandong University, 250100, Jinan, China 2 Key Laboratory of Random Complex Structures and Data Science, Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, China

Received  January 2011 Revised  March 2012 Published  March 2013

In this paper, we study the relation between the smallest $g$-supersolution of constrained backward stochastic differential equation and viscosity solution of constraint semilinear parabolic PDE, i.e. variation inequalities. And we get an existence result of variation inequalities via constrained BSDE, and prove a uniqueness result with a condition on the constraint. Then we use these results to give a probabilistic interpretation result for reflected BSDE with a discontinuous barrier and other kind of reflected BSDE.
Citation: Shige Peng, Mingyu Xu. Constrained BSDEs, viscosity solutions of variational inequalities and their applications. Mathematical Control and Related Fields, 2013, 3 (2) : 233-244. doi: 10.3934/mcrf.2013.3.233
##### References:
 [1] R. Buckdahn and Y. Hu, Hedging contingent claims for a large investor in an incomplete market, Adv. Appl. Prob., 30 (1998), 239-255. doi: 10.1239/aap/1035228002. [2] M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [3] J. Cvitanić, I. Karatzas and H. Mete Soner, Backward stochastic differential equations with constraints on the gain-process, The Annals of Probability, 26 (1998), 1522-1551. doi: 10.1214/aop/1022855872. [4] Y. Chen, M. Dai and M. Xu, Superhedging with ratio constraint, preprint, (2011). [5] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Annals of Probability, 25 (1997), 702-737. doi: 10.1214/aop/1024404416. [6] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [7] N. El Karoui and M. C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market, SIAM J. Control Optim., 33 (1995), 29-66. doi: 10.1137/S0363012992232579. [8] S. Hamadène, Reflected BSDE's with discontinuous barrier and application, Stochastics and Stochastics Reports, 74 (2002), 571-596. doi: 10.1080/1045112021000036545. [9] I. Karatzas and S. G. Kou, Hedging American contingent clains with constrained portfolios, Finance and Stochastics, 2 (1998), 215-258. doi: 10.1007/s007800050039. [10] I. Kharroub, J. Ma, H. Pham and J. Zhang, Backward SDEs with constrained jumps and quasi-variational inequalities, Annals of Probability, 38 (2010), 794-840. doi: 10.1214/09-AOP496. [11] J.-P. Lepeltier and M. Xu, Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier, Statistics and Probability Letters, 75 (2005), 58-66. doi: 10.1016/j.spl.2005.05.016. [12] S. Peng, Probabilistic interpretation for system of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, 37 (1991), 61-74. [13] S. Peng, Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Probab. Theory and Related Fields, 113 (1999), 473-499. doi: 10.1007/s004400050214. [14] S. Peng and M. Xu, Smallest $g$-Supermartingales and reflected BSDE with single and double $L^2$ obstacles, Annuals of Institute of Henri Poincaré Probab. Statist., 41 (2005), 605-630. doi: 10.1016/j.anihpb.2004.12.002. [15] S. Peng and M. Xu, $g_{\Gamma}$-expectations and the related nonlinear Doob-Meyer decomposition theorem, in "Control Theory and Related Topics," World Scientific Publishing Company, Hackensack, NJ, (2007), 122-140. doi: 10.1142/9789812790552_0010. [16] S. Peng and M. Xu, Reflected BSDE with a Constrainte and its Applications in an Incomplete Market, Bernoulli, 16 (2010), 614-640. doi: 10.3150/09-BEJ227. [17] N. Touzi, "Stochastic Control Problems, Viscosity Solutions and Application to Finance," Scuola Normale Superiore di Pisa Quaderni, Scuola Normale Superiore, Pisa, 2004.

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##### References:
 [1] R. Buckdahn and Y. Hu, Hedging contingent claims for a large investor in an incomplete market, Adv. Appl. Prob., 30 (1998), 239-255. doi: 10.1239/aap/1035228002. [2] M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [3] J. Cvitanić, I. Karatzas and H. Mete Soner, Backward stochastic differential equations with constraints on the gain-process, The Annals of Probability, 26 (1998), 1522-1551. doi: 10.1214/aop/1022855872. [4] Y. Chen, M. Dai and M. Xu, Superhedging with ratio constraint, preprint, (2011). [5] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Annals of Probability, 25 (1997), 702-737. doi: 10.1214/aop/1024404416. [6] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [7] N. El Karoui and M. C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market, SIAM J. Control Optim., 33 (1995), 29-66. doi: 10.1137/S0363012992232579. [8] S. Hamadène, Reflected BSDE's with discontinuous barrier and application, Stochastics and Stochastics Reports, 74 (2002), 571-596. doi: 10.1080/1045112021000036545. [9] I. Karatzas and S. G. Kou, Hedging American contingent clains with constrained portfolios, Finance and Stochastics, 2 (1998), 215-258. doi: 10.1007/s007800050039. [10] I. Kharroub, J. Ma, H. Pham and J. Zhang, Backward SDEs with constrained jumps and quasi-variational inequalities, Annals of Probability, 38 (2010), 794-840. doi: 10.1214/09-AOP496. [11] J.-P. Lepeltier and M. Xu, Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier, Statistics and Probability Letters, 75 (2005), 58-66. doi: 10.1016/j.spl.2005.05.016. [12] S. Peng, Probabilistic interpretation for system of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, 37 (1991), 61-74. [13] S. Peng, Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Probab. Theory and Related Fields, 113 (1999), 473-499. doi: 10.1007/s004400050214. [14] S. Peng and M. Xu, Smallest $g$-Supermartingales and reflected BSDE with single and double $L^2$ obstacles, Annuals of Institute of Henri Poincaré Probab. Statist., 41 (2005), 605-630. doi: 10.1016/j.anihpb.2004.12.002. [15] S. Peng and M. Xu, $g_{\Gamma}$-expectations and the related nonlinear Doob-Meyer decomposition theorem, in "Control Theory and Related Topics," World Scientific Publishing Company, Hackensack, NJ, (2007), 122-140. doi: 10.1142/9789812790552_0010. [16] S. Peng and M. Xu, Reflected BSDE with a Constrainte and its Applications in an Incomplete Market, Bernoulli, 16 (2010), 614-640. doi: 10.3150/09-BEJ227. [17] N. Touzi, "Stochastic Control Problems, Viscosity Solutions and Application to Finance," Scuola Normale Superiore di Pisa Quaderni, Scuola Normale Superiore, Pisa, 2004.
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