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Approximation scheme for solutions of backward stochastic differential equations via the representation theorem. (English) Zbl 1134.60349

Summary: We are interested in the approximation and simulation of solutions for the backward stochastic differential equations. We suggest two approximation schemes, and we study the \(\mathbb L^2\) induced error.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C35 Stochastic particle methods
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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References:

[1] Bally, V., An approximation scheme for BSDEs and applications to control and nonlinear PDE’s, Pitman Research Notes in Mathematics Series (1997)
[2] Bouchard, B.; Ekeland, I.; Touzi, N., On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance Stoch, 111 (2), 175-206 (2004) · Zbl 1071.60059
[3] Bouchard, B.; Touzi, N., Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 8 (1), 45-71 (2004) · Zbl 1071.60059
[4] Carriére, J. F., Valuation of the early-exercise price for option using simulations and nonparametric regression, Insurance:Mathematics and Economics, 19, 19-30 (1996) · Zbl 0894.62109 · doi:10.1016/S0167-6687(96)00004-2
[5] Chevance, D.; Rogers, L. C.G.; Talay, D., Numerical methods in finance, 232-244 (1997) · Zbl 0898.90031
[6] Cvitanic, J.; Karatzas, I., Backward stochastic differential equations with reflection and Dynkin games, The Annals of Probability, 24, 2024-2056 (1996) · Zbl 0876.60031 · doi:10.1214/aop/1041903216
[7] Douglas, J.; Ma, J.; Protter, P., Numerical methods for forward-backward stochastic differential equations, Annals of Applied Probability, 6, 940-968 (1996) · Zbl 0861.65131 · doi:10.1214/aoap/1034968235
[8] El Karoui, N.; Peng, S.; Quenez, Mc., Backward stochastic differential equations in finance, Mathematical Finance, 1-71 (1997) · Zbl 0884.90035 · doi:10.1111/1467-9965.00022
[9] Faure, O., Simulation du mouvement brownien et des diffusions (1992)
[10] Gobet, E.; Lemor, J. P.; Warin, X., A regression-based Monte-Carlo method to solve backward stochastic differential equations, Annals of Applied Probability, 15(3), 2172-2002 (2005) · Zbl 1083.60047 · doi:10.1214/105051605000000412
[11] Hamadène, S.; Lepeltier, J-P, Zero-sum stochastic differential games and BSDEs, Systems and Control letters, 24, 259-263 (1995) · Zbl 0877.93125 · doi:10.1016/0167-6911(94)00011-J
[12] Longstaff, F.; Schwartz, E., Valuing american options by simulation: a simple least squares approach, The review of Financial studies, 14(1), 113-147 (2001) · Zbl 1386.91144 · doi:10.1093/rfs/14.1.113
[13] Ma, J.; Protter, P.; Young, J., Solving forward backward stochastic differential equations explicitly: a four step scheme, Probability Theory and Related Fields, 98, 339-359 (1994) · Zbl 0794.60056 · doi:10.1007/BF01192258
[14] Ma, J.; Zhang, J., Representation theorems for backward stochastic differential equations, The Annals of Applied Probability, 12(4), 1390-1418 (2002) · Zbl 1017.60067
[15] Ma, J.; Zhang, J., Representation and regularities for solutions to BSDE’s with reflections, Stochastic Processes and their Applications, 115, 539-569 (2005) · Zbl 1076.60049 · doi:10.1016/j.spa.2004.05.010
[16] Pardoux, E.; Peng, S., Adapted solution of backward stochastic differential equations, Systems and control Letters, 14, 51-61 (1990) · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6
[17] Zhang, J., A numerical scheme for BSDE’s, The Annals of Applied Probability, 14(1), 459-488 (2004) · Zbl 1056.60067 · doi:10.1214/aoap/1075828058
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