Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 2, pp. 335-360.

In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.

DOI : https://doi.org/10.1051/m2an/2010059
Classification : 60H10,  34K28
Mots clés : backward stochastic differential equations, reflected stochastic differential equations with one barrier, numerical algorithm, numerical simulation
@article{M2AN_2011__45_2_335_0,
author = {Peng, Shige and Xu, Mingyu},
title = {Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {335--360},
publisher = {EDP-Sciences},
volume = {45},
number = {2},
year = {2011},
doi = {10.1051/m2an/2010059},
zbl = {1269.65008},
mrnumber = {2804642},
language = {en},
url = {archive.numdam.org/item/M2AN_2011__45_2_335_0/}
}
Peng, Shige; Xu, Mingyu. Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 2, pp. 335-360. doi : 10.1051/m2an/2010059. http://archive.numdam.org/item/M2AN_2011__45_2_335_0/

[1] V. Bally, An approximation scheme for BSDEs and applications to control and nonlinear PDE's, in Pitman Research Notes in Mathematics Series 364, Longman, New York (1997). | MR 1752682

[2] V. Bally and G. Pages, A quantization algorithm for solving discrete time multi-dimensional optimal stopping problems. Bernoulli 9 (2003) 1003-1049. | MR 2046816 | Zbl 1042.60021

[3] V. Bally and G. Pages, Error analysis of the quantization algorithm for obstacle problems. Stoch. Proc. Appl. 106 (2003) 1-40. | MR 1983041 | Zbl 1075.60523

[4] B. Bouchard and N. Touzi, Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equation. Stoch. Proc. Appl. 111 (2004) 175-206. | MR 2056536 | Zbl 1071.60059

[5] P. Briand, B. Delyon and J. Mémin, Donsker-type theorem for BSDEs. Elect. Comm. Probab. 6 (2001) 1-14. | MR 1817885 | Zbl 0977.60067

[6] P. Briand, B. Delyon and J. Mémin, On the robustness of backward stochastic differential equations. Stoch. Process. Appl. 97 (2002) 229-253. | MR 1875334 | Zbl 1058.60041

[7] D. Chevance, Résolution numérique des équations différentielles stochastiques rétrogrades, in Numerical Methods in Finance, Cambridge University Press, Cambridge (1997). | Zbl 0898.90031

[8] F. Coquet, V. Mackevicius and J. Mémin, Stability in D of martingales and backward equations under discretization of filtration. Stoch. Process. Appl. 75 (1998) 235-248. | MR 1632205 | Zbl 0932.60047

[9] J. Cvitanic, I. Karatzas and M. Soner, Backward stochastic differential equations with constraints on the gain-process. Ann. Probab. 26 (1998) 1522-1551. | MR 1675035 | Zbl 0935.60039

[10] F. Delarue and S. Menozzi, An interpolated Stochastic Algorithm for Quasi-Linear PDEs. Math. Comput. 261 (2008) 125-158. | MR 2353946 | Zbl 1131.65002

[11] J. Douglas, J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab. 6 (1996) 940-968. | MR 1410123 | Zbl 0861.65131

[12] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M.-C. Quenez, Reflected solutions of backward SDE and related obstacle problems for PDEs. Ann. Probab. 25 (1997) 702-737. | MR 1434123 | Zbl 0899.60047

[13] N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1-71. | MR 1434407 | Zbl 0884.90035

[14] E. Gobet, J.P. Lemor and X. Warin, Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12 (2006) 889-916. | MR 2265667 | Zbl 1136.60351

[15] P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992). | MR 1214374 | Zbl 1216.60052

[16] J. Ma, P. Protter, J. San Martín and S. Torres, Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12 (2002) 302-316. | MR 1890066 | Zbl 1017.60074

[17] J. Mémin, S. Peng and M. Xu, Convergence of solutions of discrete reflected backward SDE's and simulations. Acta Math. Appl. Sin. (English Series) 24 (2008) 1-18. | MR 2385005 | Zbl 1138.60049

[18] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14 (1990) 55-61. | MR 1037747 | Zbl 0692.93064

[19] S. Peng, Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probab. Theory Relat. Fields 113 (1999) 473-499. | MR 1717527 | Zbl 0953.60059

[20] S. Peng and M. Xu, Reflected BSDE with Constraints and the Related Nonlinear Doob-Meyer Decomposition. Preprint, available at e-print:arXiv:math/0611869v4 (2006).

[21] E.G. Rosazza, Risk measures via $g$-expectations. Insur. Math. Econ. 39 (2006) 19-34. | MR 2241848 | Zbl 1147.91346

[22] M. Xu, Numerical algorithms and simulations for reflected BSDE with two barriers. Preprint, available at arXiv:0803.3712v2 [math.PR] (2007).

[23] J. Zhang, Some fine properties of backward stochastic differential equations. Ph.D. Thesis, Purdue University (2001). | MR 2703162

[24] J. Zhang, A numerical scheme for BSDEs. Ann. Appl. Probab. 14 (2004) 459-488. | MR 2023027 | Zbl 1056.60067

[25] Y. Zhang and W. Zheng, Discretizing a backward stochastic differential equation. Int. J. Math. Math. Sci. 32 (2002) 103-116. | MR 1937828 | Zbl 1006.60050