Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations

Peng, Shige; Xu, Mingyu

ESAIM: Mathematical Modelling and Numerical Analysis , 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.

Classification : 60H10, 34K28
Mots clés : backward stochastic differential equations, reflected stochastic differential equations with one barrier, numerical algorithm, numerical simulation

[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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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

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

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