The Annals of Applied Probability

Numerical simulation of BSDEs with drivers of quadratic growth

Adrien Richou

Full-text: Open access


This article deals with the numerical resolution of Markovian backward stochastic differential equations (BSDEs) with drivers of quadratic growth with respect to z and bounded terminal conditions. We first show some bound estimates on the process Z and we specify the Zhang’s path regularity theorem. Then we give a new time discretization scheme with a nonuniform time net for such BSDEs and we obtain an explicit convergence rate for this scheme.

Article information

Ann. Appl. Probab. Volume 21, Number 5 (2011), 1933-1964.

First available in Project Euclid: 25 October 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 65C30: Stochastic differential and integral equations 60H10: Stochastic ordinary differential equations [See also 34F05]

BSDEs driver of quadratic growth time discretization scheme


Richou, Adrien. Numerical simulation of BSDEs with drivers of quadratic growth. Ann. Appl. Probab. 21 (2011), no. 5, 1933--1964. doi:10.1214/10-AAP744.

Export citation


  • [1] Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
  • [2] Briand, P. and Confortola, F. (2008). BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces. Stochastic Process. Appl. 118 818–838.
  • [3] Briand, P. and Hu, Y. (2006). BSDE with quadratic growth and unbounded terminal value. Probab. Theory Related Fields 136 604–618.
  • [4] Briand, P. and Hu, Y. (2008). Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 543–567.
  • [5] Cheridito, P. and Stadje, M. (2010). BSΔEs and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness. Available at arXiv:1002.0175v1.
  • [6] Chevance, D. (1997). Numerical methods for backward stochastic differential equations. In Numerical Methods in Finance. Publ. Newton Inst. 232–244. Cambridge Univ. Press, Cambridge.
  • [7] Delarue, F. and Menozzi, S. (2006). A forward–backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab. 16 140–184.
  • [8] Delbaen, F., Hu, Y. and Bao, X. (2011). Backward SDEs with superquadratic growth. Probab. Theory Related Fields 150 145–192.
  • [9] Delbaen, F., Hu, Y. and Richou, A. (2011). On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • [10] Fuhrman, M. and Tessitore, G. (2002). The Bismut–Elworthy formula for backward SDEs and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces. Stochastics Stochastics Rep. 74 429–464.
  • [11] 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.
  • [12] Gobet, E. and Makhlouf, A. (2010). L2-time regularity of BSDEs with irregular terminal functions. Stochastic Process. Appl. 120 1105–1132.
  • [13] Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Probab. 15 1691–1712.
  • [14] Imkeller, P. and Dos Reis, G. (2010). Path regularity and explicit convergence rate for BSDE with truncated quadratic growth. Stochastic Process. Appl. 120 348–379.
  • [15] Imkeller, P., dos Reis, G. and Zhang, J. (2010). Results on numerics for FBSDE with drivers of quadratic growth. In Contemporary Quantitative Finance (A. Chiarella and C. Novikov, eds.). Springer, Berlin.
  • [16] Kazamaki, N. (1994). Continuous Exponential Martingales and BMO. Lecture Notes in Math. 1579. Springer, Berlin.
  • [17] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
  • [18] Lepeltier, J. P. and San Martín, J. (1998). Existence for BSDE with superlinear-quadratic coefficient. Stochastics Stochastics Rep. 63 227–240.
  • [19] Zhang, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459–488.
  • [20] Zhang, J. (2005). Representation of solutions to BSDEs associated with a degenerate FSDE. Ann. Appl. Probab. 15 1798–1831.
  • [21] Zhang, J. (2001). Some fine properties of BSDE. Ph.D. thesis, Purdue Univ.