Backward SDEs with constrained jumps and quasi-variational inequalities.(English)Zbl 1205.60114

The authors give solutions to backward stochastic differential equations (BSDEs) driven by Brownian motion and a Poisson random measure, and subject to constraints on the jump component. Some impulse control problems are the motivation for investigation of such BSDEs. A problem with constraints is considered in this paper, so we may not have the uniqueness of the solution. Therefore the authors look for the minimal solution using a penalization approach. They show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs) which is the dynamic programming equation associated to the impulse control problems. Moreover, the authors give a probabilistic representation for solutions to QVIs and a new stochastic formula for value functions of a class of impulse control problems. This also suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs. As a direct consequence, this suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs. Such BSDEs are also useful in problems of mathematical finance, namely in problems of hedging of payoffs.

MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.) 35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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 [1] Barles, G. (1994). Solutions de Viscosité des équations de Hamilton-Jacobi. Mathématiques et Applications 17 . Springer, Paris. · Zbl 0819.35002 [2] Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60 57-83. · Zbl 0878.60036 [3] Bensoussan, A. and Lions, J.-L. (1984). Impulse Control and Quasivariational Inequalities . Gauthier-Villars, Montrouge. [4] Bouchard, B. (2009). A stochastic target formulation for optimal switching problems in finite horizon. Stochastics 81 171-197. · Zbl 1175.60037 [5] Buckdahn, R. and Hu, Y. (1998). Hedging contingent claims for a large investor in an incomplete market. Adv. in Appl. Probab. 30 239-255. · Zbl 0904.90009 [6] Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. ( N.S. ) 27 1-67. · Zbl 0755.35015 [7] Cvitanić, J., Karatzas, I. and Soner, H. M. (1998). Backward stochastic differential equations with constraints on the gains-process. Ann. Probab. 26 1522-1551. · Zbl 0935.60039 [8] Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel. I-IV. Hermann, Paris. · Zbl 0323.60039 [9] Dellacherie, C. and Meyer, P.-A. (1980). Probabilités et Potentiel. V-VIII. Hermann, Paris. · Zbl 0464.60001 [10] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 702-737. · Zbl 0899.60047 [11] El Karoui, N. and Quenez, M.-C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29-66. · Zbl 0831.90010 [12] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions , 2nd ed. Stochastic Modelling and Applied Probability 25 . Springer, New York. · Zbl 1105.60005 [13] Hamadène, S. and Ouknine, Y. (2003). Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Probab. 8 1-20. · Zbl 1015.60051 [14] Ishii, K. (1993). Viscosity solutions of nonlinear second order elliptic PDEs associated with impulse control problems. Funkcial. Ekvac. 36 123-141. · Zbl 0831.49024 [15] Ma, J. and Zhang, J. (2002). Path regularity for solutions of backward stochastic differential equations. Probab. Theory Related Fields 122 163-190. · Zbl 1014.60060 [16] Meyer, P.-A. and Zheng, W. A. (1984). Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 20 353-372. · Zbl 0551.60046 [17] Øksendal, B. and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions , 2nd ed. Springer, Berlin. · Zbl 1116.93004 [18] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. Probab. Theory Related Fields 113 473-499. · Zbl 0953.60059 [19] Peng, S. and Xu, M. (2007). Constrained BSDE and viscosity solutions of variation inequalities. [20] Royer, M. (2006). Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process. Appl. 116 1358-1376. · Zbl 1110.60062 [21] Tang, S. J. and Li, X. J. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 1447-1475. · Zbl 0922.49021 [22] Tang, S. J. and Yong, J. M. (1993). Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach. Stochastics Stochastics Rep. 45 145-176. · Zbl 0795.93103
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