Euler-type schemes for weakly coupled forward-backward stochastic differential equations and optimal convergence analysis

Abstract

We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations (FBSDEs). Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18: 143–177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.

Key words： Weakly coupled forward-backward stochastic differential equations (FBSDEs); Euler-type scheme; time discretization; first-order; error estimate

References

1 Bally V. Approximation scheme for solutions of BSDE. In: Backward Stochastic Differential Equations (Paris, 1995-1996), Pitman Res Notes Math, Ser 364. Harlow: Longman, 1997: 177-191

2 Bally V, Pages G. A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli, 2003, 9: 1003-1049

3 Bender C, Zhang J. Time discretization and Markovian iteration for coupled FBSDEs. Ann Appl Probab, 2008, 18: 143-177

4 Bouchard B, Touzi N. Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process Appl, 2004, 111: 175-206

5 Crisan D, Manolarakis K. Second order discretization of backward S<?Pub Caret?>DEs. arXiv: 1012.5650

6 Cvitanic J, Zhang J. The steepest descent method for forward-backward SDEs. Electron J Probab, 2005, 10: 1468-1495

7 Delarue F, Menozzi S. A forward-backward stochastic algorithm for quasi-linear PDEs. Ann Appl Probab, 2006, 16: 140-184

8 Delarue F, Menozzi S. An interpolated stochastic algorithm for quasi-linear PDEs. Math Comp, 2008, 77: 125-158

9 Douglas J, Ma J, Protter P. Numerical methods for forward-backward stochastic differential equations. Ann Appl Probab, 1996, 6: 940-968

10 Gianin E R. Risk measures via g-expectations. Insurance Math Econom, 2006, 39: 19-34

11 Hamadene S, Lepeltier J P. Zero-sum stochastic differential games and backward equations. Systems Control Lett, 1995, 24: 259-263

12 Karoui N EL, Peng S G, Quenez M C. Backward stochastic differential equations in finance. Math Finance, 1997, 7: 1-71

13 Lemor J P, Gobet E, Warin X. A regression-based Monte Carlo method for backward stochastic differential equations. Ann Appl Probab, 2005, 15: 2172-2202

14 Lemor J P, Gobet E, Warin X. Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli, 2006, 12: 889-916

15 Li Y, Zhao W D. L_p-error estimates for numerical schemes for solving certain kinds of backward stochastic differential equations. Statist Probab Lett, 2010, 80: 1612-1617

16 Ma J, Protter P, Martin J S, Torres S. Numerical method for backward stochastic differential equations. Ann Appl Probab, 2002, 12: 302-316

17 Ma J, Protter P, Yong J M. Solving forward-backward stochastic differential equations explicitly—a four step scheme. Probab Theory Related Fields, 1994, 98: 339-359

18 Ma J, Shen J, Zhao Y. On numerical approximations of forward-backward stochastic differential equations. SIAM J Numer Anal, 2008, 46: 2636-2661

19 Ma J, Yong J M. Forward-Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Mathematics, Vol 1702. Berlin: Springer, 1999

20 Milstein G N, Tretyakov M V. Numerical algorithms for forward-backward stochastic differential equations. SIAM J Sci Comput, 2006, 28: 561-582

21 Pardoux E, Peng S G. Adapted solution of a backward stochastic differential equation. Systems Control Lett, 1990, 14: 55-61

22 Pardoux E, Tang S. Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab Theory Related Fields, 1999, 114: 123-150

23 Peng S G. A general stochastic maximum principle for optimal control problems. SIAM J Control Optim, 1990, 28: 966-979

24 Peng S G. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics and Stochastic Reports, 1991, 37: 61-74

25 Peng S G. Backward SDE and related g-Expectation (Paris, 1995-1996). Pitman Res Notes Math, Ser 364. Harlow: Longman, 1997: 141-159

26 Peng S G. Nonlinear expectations, nonlinear evaluations and risk measures. In: Stochastic Methods in Finance. Lecture Notes in Mathematics, Vol 1856. 2004, 243-256

27 Peng S G, Wu Z. Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J Control Optim, 1999, 37: 825-843

28 Zhang J. A numerical scheme for BSDEs. Ann Appl Probab, 2004, 14: 459-488

29 Zhao W D, Chen L F, Peng S G. A new kind of accurate numerical method for backward stochastic differential equations. SIAM J Sci Comput, 2006, 28: 1563-1581

30 Zhao W D, Fu Y, Zhou T. A new kind of high-order multi-step schemes for forward backward stochastic differential equations. arXiv: 1310.5307, revised

31 Zhao W D, Li Y, Ju L L. Error estimates of the Crank-Nicolson scheme for solving backward stochastic differential equations. Int J Numer Anal Model, 2013, 10: 876-898

32 Zhao W D, Li Y, Zhang G N. A generalized θ-scheme for solving backward stochastic differential equations. Discrete Contin Dyn Syst Ser B, 2012, 17: 1585-1603

33 Zhao W D, Wang J L, Peng S G. Error estimates of the θ-scheme for backward stochastic differential equations. Discrete Contin Dyn Syst Ser B, 2009, 12: 905-924

34 Zhao W D, Zhang G N, Ju L L. A stable multistep scheme for solving backward stochastic differential equations. SIAM J Numer Anal, 2010, 48: 1369-1394

35 Zhao W D, Zhang W, Ju L L. A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations. Commun Comput Phys, 2014, 15: 618-646