Time discretization and Markovian iteration for coupled FBSDEs

Annals of Applied Probability

Volumn 18, Issue 1, 2008, Pages 143-177

  Bender, Christian ^a   Zhang, Jianfeng ^a,b  

^a TECHNICAL UNIVERSITY OF BRAUNSCHWEIG   (Germany)

^b UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

Forward backward SDE; Monte Carlo simulation; Numerics; Time discretization

Indexed keywords

EID: 53349147977     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/07-AAP448     Document Type: Article

References (26)

1. ANTONELLI, A. (1993). Backward-forward stochastic differential equations. Ann. Appl. Probab. 3 777-793. MR1233625
2. BALLY, V. and PAGÈS, G. (2003). Error analysis of the quantization algorithm for obstacle problems. Stochastic Process. Appl. 106 1-40. MR1983041
3. BENDER, C. and DENK, R. (2007). A forward scheme for backward SDEs. Stochastic Process. Appl. To appear.
4. BOUCHARD, B. and TOUZI, N. (2004). Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175-206. MR2056536
5. BRIGO, D. and MERCURIO, F. (2004). Interest Rate Models: Theory and Practice. Springer, Berlin. MR2255741
6. CARRIÈRE, J. F. (1996). Valuation of the early-exercise price for options using simulations and nonparametric regression. Insurance Math. Econom. 19 19-30. MR1439613
7. CLÉMENT, E., LAMBERTON, D. and PROTTER, P. (2002). An analysis of a least squares regression algorithm for American option pricing. Finance Stoch. 6 449-471. MR1932380
8. CVITANIĆ, J. and ZHANG, J. (2005). The steepest descent method for FBSDEs. Electron. J. Probab. 10 1468-1495. MR2191636
9. DELARUE, F. (2002). On the existence and uniqueness of solutions to FBSDEs in a nondegenerate case. Stochastic Process. Appl. 99 209-286. MR1901154
10. DELARUE, F. and MENOZZI, S. (2006). A forward-backward stochastic algorithm for quasilinear PDEs. Ann. Appl. Probab. 16 140-184. MR2209339
11. DOUGLAS, J., MA, J. and PROTTER, P. (1996). Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab. 6 940-968. MR1410123
12. EL KAROUI, N., PENG, S. and QUENEZ, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71. MR1434407
13. GLASSERMAN, P. (2003). Monte Carlo Methods in Financial Engineering. Springer, New York. MR1999614
14. GOBET, E., LEMOR, J.-P. and WARIN, X. (2005). A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15 2172-2202. MR2152657
15. HULL, J. and WHITE, A. (1990). Pricing interest rate securities. Rev. Financial Studies 3 573-592.
16. LONGSTAFF, F. A. and SCHWARTZ, R. S. (2001). Valuing American options by simulation: A simple least-squares approach. Rev. Financial Studies 14 113-147.
17. MA, J., PROTTER, P. and YONG, J. (1994). Solving forward-backward stochastic differential equations explicitly - a four step scheme. Probab. Theory Related Fields 98 339-359. MR1262970
18. MA, J., SHEN, J. and ZHAO, Y. (2006). Numerical method for forward backward SDEs: Four step scheme revisited. Preprint.
19. MA, J. and YONG, J. (1999). Forward Backward Stochastic Differential Equations and Their Applications. Springer, Berlin. MR1704232
20. MILSTEIN, G. N. and TRETYAKOV, M. V. (2006). Numerical algorithms for forward-backward stochastic differential equations. SIAM J. Sci. Comput. 28 561-582. MR2231721
21. PARDOUX, E. and PENG, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. Lecture Notes in Control and Inform. Sci. 176 200-217. Springer, Berlin. MR1176785
22. PARDOUX, E. and TANG, S. (1999). Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Related Fields 114 123-150. MR1701517
23. RIVIERE, O. (2005). Equations différentielles stochastiques progressives rétrogrades couplées: Equations aux dérivées partielles et discrétisation. Ph.D. thesis, Paris.
24. ZHANG, J. (2001). Some fine properties of backward stochastic differential equations. Ph.D. dissertation, Purdue Univ.
25. ZHANG, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459-488. MR2023027
26. ZHANG, J. (2006). The wellposedness of FBSDEs. Discrete Contin. Dyn. Syst. Ser. B 6 927-940. MR2223916