Electronic Journal of Probability

The Steepest Descent Method for Forward-Backward SDEs

Jaksa Cvitanic and Jianfeng Zhang

Full-text: Open access

Abstract

This paper aims to open a door to Monte-Carlo methods for numerically solving Forward-Backward SDEs, without computing over all Cartesian grids as usually done in the literature. We transform the FBSDE to a control problem and propose the steepest descent method to solve the latter one. We show that the original (coupled) FBSDE can be approximated by decoupled FBSDEs, which further comes down to computing a sequence of conditional expectations. The rate of convergence is obtained, and the key to its proof is a new well-posedness result for FBSDEs. However, the approximating decoupled FBSDEs are non-Markovian. Some Markovian type of modification is needed in order to make the algorithm efficiently implementable.

Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 45, 1468-1495.

Dates
Accepted: 19 December 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816846

Digital Object Identifier
doi:10.1214/EJP.v10-295

Mathematical Reviews number (MathSciNet)
MR2191636

Zentralblatt MATH identifier
1109.60056

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Cvitanic, Jaksa; Zhang, Jianfeng. The Steepest Descent Method for Forward-Backward SDEs. Electron. J. Probab. 10 (2005), paper no. 45, 1468--1495. doi:10.1214/EJP.v10-295. https://projecteuclid.org/euclid.ejp/1464816846


Export citation

References

  • Bally, V. Approximation scheme for solutions of BSDE. Backward stochastic differential equations (Paris, 1995–1996), 177–191, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997.
  • Bally, Vlad; Pagès, Gilles; Printems, Jacques. A quantization tree method for pricing and hedging multidimensional American options. Math. Finance 15 (2005), no. 1, 119–168.
  • Bender, C. and Denk R. Forward Simulation of Backward SDEs, preprint
  • Bouchard, Bruno; Touzi, Nizar. Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 (2004), no. 2, 175–206.
  • Briand, Philippe; Delyon, Bernard; Mémin, Jean. Donsker-type theorem for BSDEs. Electron. Comm. Probab. 6 (2001), 1–14 (electronic).
  • Chevance, D. Numerical methods for backward stochastic differential equations. Numerical methods in finance, 232–244, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997.
  • Delarue, François. On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stochastic Process. Appl. 99 (2002), no. 2, 209–286.
  • Delarue F. and Menozzi S. A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab., to appear.
  • Douglas, Jim, Jr.; Ma, Jin; Protter, Philip. Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab. 6 (1996), no. 3, 940–968.
  • Gobet, Emmanuel; Lemor, Jean-Philippe; Warin, Xavier. A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15 (2005), no. 3, 2172–2202.
  • Ladyzhenskaja, O. A.; Solonnikov, V. A.; Uralcprime ceva, N. N. Linear and quasilinear equations of parabolic type. (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967 xi+648 pp. (39 #3159b)
  • Ma, Jin; Protter, Philip; San Martín, Jaime; Torres, Soledad. Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12 (2002), no. 1, 302–316.
  • Ma, Jin; Protter, Philip; Yong, Jiong Min. Solving forward-backward stochastic differential equations explicitly–-a four step scheme. Probab. Theory Related Fields 98 (1994), no. 3, 339–359.
  • Makarov, R. N. Numerical solution of quasilinear parabolic equations and backward stochastic differential equations. Russian J. Numer. Anal. Math. Modelling 18 (2003), no. 5, 397–412.
  • Mémin J., Peng S., and Xu M. Convergence of solutions of discrete reflected BSDEs and simulations, preprint.
  • Milstein, G. N.; Tretyakov, M. V.. Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations. Math. Comp. 69 (2000), no. 229, 237–267.
  • Ma, Jin; Yong, Jiongmin. Forward-backward stochastic differential equations and their applications. Lecture Notes in Mathematics, 1702. Springer-Verlag, Berlin, 1999. xiv+270 pp. ISBN: 3-540-65960-9
  • Ma, J.; Yong, J.. Approximate solvability of forward-backward stochastic differential equations. Appl. Math. Optim. 45 (2002), no. 1, 1–22.
  • Pardoux, É.; Peng, S. G.. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55–61.
  • Pardoux, Etienne; Tang, Shanjian. Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Related Fields 114 (1999), no. 2, 123–150.
  • Zhang, Jianfeng. A numerical scheme for BSDEs. Ann. Appl. Probab. 14 (2004), no. 1, 459–488.
  • Zhang J. The well-posedness of FBSDEs, Discrete and Continuous Dynamical Systems (B), to appear
  • Zhang J. The well-posedness of FBSDEs (II), Submitted
  • Zhang, Yinnan; Zheng, Weian. Discretizing a backward stochastic differential equation. Int. J. Math. Math. Sci. 32 (2002), no. 2, 103–116.
  • Zhou, Xun Yu. Stochastic near-optimal controls: necessary and sufficient conditions for near-optimality. SIAM J. Control Optim. 36 (1998), no. 3, 929–947 (electronic).