Bernoulli

  • Bernoulli
  • Volume 15, Number 4 (2009), 1117-1147.

Strong approximations of BSDEs in a domain

Bruno Bouchard and Stéphane Menozzi

Full-text: Open access

Abstract

We study the strong approximation of a backward SDE with finite stopping time horizon, namely the first exit time of a forward SDE from a cylindrical domain. We use the Euler scheme approach of (Stochastic Process. Appl. 111 (2004) 175–206 and Ann. Appl. Probab. 14 (2004) 459–488). When the domain is piecewise smooth and under a non-characteristic boundary condition, we show that the associated strong error is at most of order h1/4−ɛ, where h denotes the time step and ɛ is any positive parameter. This rate corresponds to the strong exit time approximation. It is improved to h1/2−ɛ when the exit time can be exactly simulated or for a weaker form of the approximation error. Importantly, these results are obtained without uniform ellipticity condition.

Article information

Source
Bernoulli, Volume 15, Number 4 (2009), 1117-1147.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1262962228

Digital Object Identifier
doi:10.3150/08-BEJ181

Mathematical Reviews number (MathSciNet)
MR2597585

Zentralblatt MATH identifier
1204.60048

Keywords
backward SDEs discrete-time approximation first boundary value problem

Citation

Bouchard, Bruno; Menozzi, Stéphane. Strong approximations of BSDEs in a domain. Bernoulli 15 (2009), no. 4, 1117--1147. doi:10.3150/08-BEJ181. https://projecteuclid.org/euclid.bj/1262962228


Export citation

References

  • [1] Bally, V. and Pages, G. (2002). A quantization algorithm for solving discrete time multidimensional optimal stopping problems. Bernoulli 9 1003–1049.
  • [2] Bouchard, B. and Chassagneux, J.-F. (2008). Discrete-time approximation for continuously and discretely reflected BSDE’s. Stochastic Process. Appl. 118 2269–2293.
  • [3] Bouchard, B. and Elie, R. (2005). Discrete time approximation of decoupled forward–backward SDE with jumps. Stochastic Process. Appl. 118 53–75.
  • [4] Bouchard, B. and Menozzi, S. (2008). Strong approximation of BSDEs in a domain. Available at http://hal.archives-ouvertes.fr/hal-00177481/fr/.
  • [5] Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
  • [6] Crandall, M.G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Amer. Math. Soc. 27 1–67.
  • [7] Darling, R.W.R. and Pardoux, E. (1997). BSDE with random terminal time. Ann. Probab. 25 1135–1159.
  • [8] Delarue, F. and Menozzi, S. (2006). A forward–backward algorithm for quasi-linear PDEs. Ann. Appl. Probab. 16 140–184.
  • [9] Delarue, F. and Menozzi, S. (2008). An interpolated stochastic algorithm for quasi-linear PDEs. Math. Comp. 77 125–158.
  • [10] Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Annals of Mathematics Studies 109. Princeton, NJ: Princeton Univ. Press.
  • [11] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Englewood Cliffs, NJ: Prentice Hall.
  • [12] Gilbarg, D. and Trudinger, N.S. (1998). Elliptic Partial Differential Equations of Second Order. Reprint of 1998 edition. Classics in Mathematics. Berlin: Springer, 2001. xiv+517.
  • [13] Gobet, E. (1998). Schéma d’Euler pour diffusions tuées. Application aux options barriére. Ph.D. thesis, Univ. Paris VII.
  • [14] Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes. Stochastic Process. Appl. 87 167–197.
  • [15] Gobet, E. and Labart, C. (2007). Error expansion for the discretization of backward stochastic differential equations. Stochastic Process. Appl. 117 803–829.
  • [16] Gobet, E., Lemor, J.P. and Warin, X. (2006). Rate of convergence of empirical regression method for solving generalized BSDE. Bernoulli 12 889–916.
  • [17] Gobet, E. and Menozzi, S. (2007). Stopped diffusion processes: Overshoots and boundary correction. Preprint PMA, Univ. Paris 7.
  • [18] Gobet, E. and Menozzi, S. (2004). Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme. Stochastic Process. Appl. 114 201–223.
  • [19] Karatzas, I. and Shreve, S.E. (1990). Brownian Motion and Stochastic Calculus. New York: Springer.
  • [20] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
  • [21] Lieberman, G.M. (2005). Second Order Parabolic Differential Equations. River Edge, NJ: World Scientific.
  • [22] Longstaff, F.A. and Schwartz, R.S. (2001). Valuing American options by simulation: A simple least-square approach. Rev. Financial Stud. 14 113–147.
  • [23] Ma, J. and Zhang, J. (2002). Path regularity of solutions to backward stochastic differential equations. Probab. Theory Related Fields 122 163–190.
  • [24] Ma, J. and Zhang, J. (2005). Representations and regularities for solutions to BSDEs with reflections. Stochastic Process. Appl. 115 539–569.
  • [25] Milstein, G.N. and Tretyakov, M.V. (2001). Numerical solution of Dirichlet problems for nonlinear parabolic equations by a probabilistic approach. IMA J. Num. Anal. 21 887–917.
  • [26] Pardoux, E. (1998). Backward stochastic differential equations and viscosity solutions of semilinear parabolic and elliptic PDE’s of second order. In Stochastic Analysis and Related Topics: The Geilo Workshop 1996 (L. Decreusefond, J. Gjerd, B. Oksendal and A.S. Ustünel, eds.) 79–127. Boston: Birkhäuser.
  • [27] 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.
  • [28] Peng, S. (1991). Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep. 37 61–74.
  • [29] Zhang, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459–488.