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Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles. (English) Zbl 1342.60113

Summary: We study a discrete time approximation scheme for the solution of a doubly reflected backward stochastic differential equation (DBBSDE in short) with jumps, driven by a Brownian motion and an independent compensated Poisson process. Moreover, we suppose that the obstacles are right continuous and left limited (RCLL) processes with predictable and totally inaccessible jumps and satisfy Mokobodzki’s condition. Our main contribution consists in the construction of an implementable numerical scheme, based on two random binomial trees and the penalization method, which is shown to converge to the solution of the DBBSDE. Finally, we illustrate the theoretical results with some numerical examples in the case of general jumps.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
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References:

[1] Bouchard, B.; Elie, R., Discrete-time approximation of decoupled forward-backward SDE with jumps, Stochastic Process. Appl., 118, 53-75 (2008) · Zbl 1136.60048
[2] Briand, P.; Delyon, B.; Mémin, J., Donsker-type theorem for BSDEs, Electron. Commun. Probab., 6, 1-14 (2001) · Zbl 0977.60067
[3] Chassagneux, J.-F., A discrete-time approximation for doubly reflected BSDEs, Adv. in Appl. Probab., 41, 1, 101-130 (2009) · Zbl 1166.65001
[4] Crépey, S.; Matoussi, A., Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison, Ann. Appl. Probab., 18, 5, 2041-2069 (2008) · Zbl 1158.60021
[5] Cvitanic, J.; Karatzas, I., Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 41, 2024-2056 (1996) · Zbl 0876.60031
[6] Dumitrescu, R.; Quenez, M.; Sulem, A., Generalized Dynkin games and doubly reflected BSDEs with jumps (2014) · Zbl 1351.93170
[7] El Karoui, N.; Kapoudjian, C.; Pardoux, E.; Peng, S.; Quenez, M., Reflected solutions of backward SDE’s and related obstacle problems for PDE’s, Ann. Probab., 25, 2, 702-737 (1997) · Zbl 0899.60047
[8] Essaky, E., Reflected backward stochastic differential equation with jumps and RCLL obstacle, Bull. Sci. Math., 132, 690-710 (2008) · Zbl 1157.60057
[9] Essaky, E.; Harraj, N.; Ouknine, Y., Backward stochastic differential equation with two reflecting barriers and jumps, Stoch. Anal. Appl., 23, 921-938 (2005) · Zbl 1082.60049
[10] Hamadène, S.; Hassani, M., BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game, Electron. J. Probab., 11, 121-145 (2006) · Zbl 1184.91038
[11] Hamadène, S.; Ouknine, Y., Reflected backward stochastic differential equation with jumps and random obstable, Electron. J. Probab., 8, 1-20 (2003) · Zbl 1015.60051
[12] Hamadène, S.; Ouknine, Y., Reflected backward SDEs with general jumps (2013) · Zbl 1341.60054
[13] Hamadène, S.; Wang, H., BSDEs with two RCLL reflecting obstacles driven by a Brownian motion and poisson measure and related mixed zero-sum games, Stochastic Process. Appl., 119, 2881-2912 (2009) · Zbl 1229.60083
[14] Jakubowski, A.; Mémin, J.; Pagès, G., Convergence en loi des suites d’intégrales stochastiques sur l’espace \(D^1\) de Skorokhod, Probab. Theory Related Fields, 81, 111-137 (1989) · Zbl 0638.60049
[15] Kifer, Y., Dynkin’s games and Israeli options, ISRN Probab. Stat., 2013, Article 856458 pp. (2013) · Zbl 1271.91022
[16] Lejay, A.; Mordecki, E.; Torres, S., Numerical approximation of backward stochastic differential equations with jumps (2014)
[17] Lepeltier, J.; Xu, M., Reflected backward stochastic differential equations with two RCLL barriers, ESAIM Probab. Stat., 11, 3-22 (2007) · Zbl 1171.60352
[18] Mémin, J.; Peng, S.; Xu, M., Convergence of solutions of discrete reflected backward SDE’s and simulations, Acta Math. Sin., 24, 1, 1-18 (2002) · Zbl 1138.60049
[19] Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14, 1, 55-61 (1990) · Zbl 0692.93064
[20] Peng, S.; Xu, M., The smallest g-supermartingale and reflected BSDE with single and double \(L^2\) obstacles, Ann. Inst. Henri Poincaré, 41, 605-630 (2005) · Zbl 1071.60049
[21] Peng, S.; Xu, M., Numerical algorithms for BSDEs with 1-d Brownian motion: convergence and simulation, ESAIM Math. Model. Numer. Anal., 45, 335-360 (2011) · Zbl 1269.65008
[22] Protter, P., Stochastic Integration and Differential Equations, a New Approach, Appl. Math., vol. 21 (2005), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York
[23] Quenez, M.; Sulem, A., BSDEs with jumps, optimization and applications to dynamic risk measures, Stochastic Process. Appl., 123, 3328-3357 (2013) · Zbl 1285.93091
[24] Quenez, M.; Sulem, A., Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stochastic Process. Appl., 124, 9, 3031-3054 (2014) · Zbl 1293.93783
[25] Royer, M., Backward stochastic differential equations with jumps and related non-linear expectations, Stochastic Process. Appl., 116, 10, 1358-1376 (2006) · Zbl 1110.60062
[26] Słomiński, L., Stability of strong solutions of stochastic differential equations, Stochastic Process. Appl., 31, 2, 173-202 (1989) · Zbl 0673.60065
[27] Tang, S.; Li, X., Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32, 1447-1475 (1994) · Zbl 0922.49021
[28] Xu, M., Numerical algorithms and simulations for reflected backward stochastic differential equations with two continuous barriers, J. Comput. Appl. Math., 236, 1137-1154 (2011) · Zbl 1243.65013
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