## The Annals of Applied Probability

### Time discretization of FBSDE with polynomial growth drivers and reaction–diffusion PDEs

#### Abstract

In this paper, we undertake the error analysis of the time discretization of systems of Forward–Backward Stochastic Differential Equations (FBSDEs) with drivers having polynomial growth and that are also monotone in the state variable.

We show with a counter-example that the natural explicit Euler scheme may diverge, unlike in the canonical Lipschitz driver case. This is due to the lack of a certain stability property of the Euler scheme which is essential to obtain convergence. However, a thorough analysis of the family of $\theta$-schemes reveals that this required stability property can be recovered if the scheme is sufficiently implicit. As a by-product of our analysis, we shed some light on higher order approximation schemes for FBSDEs under non-Lipschitz condition. We then return to fully explicit schemes and show that an appropriately tamed version of the explicit Euler scheme enjoys the required stability property and as a consequence converges.

In order to establish convergence of the several discretizations, we extend the canonical path- and first-order variational regularity results to FBSDEs with polynomial growth drivers which are also monotone. These results are of independent interest for the theory of FBSDEs.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 5 (2015), 2563-2625.

Dates
Revised: June 2014
First available in Project Euclid: 30 July 2015

https://projecteuclid.org/euclid.aoap/1438261049

Digital Object Identifier
doi:10.1214/14-AAP1056

Mathematical Reviews number (MathSciNet)
MR3375884

Zentralblatt MATH identifier
1342.65011

#### Citation

Lionnet, Arnaud; dos Reis, Gonçalo; Szpruch, Lukasz. Time discretization of FBSDE with polynomial growth drivers and reaction–diffusion PDEs. Ann. Appl. Probab. 25 (2015), no. 5, 2563--2625. doi:10.1214/14-AAP1056. https://projecteuclid.org/euclid.aoap/1438261049

#### References

• Alanko, S. and Avellaneda, M. (2013). Reducing variance in the numerical solution of BSDEs. C. R. Math. Acad. Sci. Paris 351 135–138.
• Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
• Briand, P. and Carmona, R. (2000). BSDEs with polynomial growth generators. J. Appl. Math. Stoch. Anal. 13 207–238.
• Briand, P. and Confortola, F. (2008). Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators. Appl. Math. Optim. 57 149–176.
• Briand, Ph., Delyon, B., Hu, Y., Pardoux, E. and Stoica, L. (2003). $L^{p}$ solutions of backward stochastic differential equations. Stochastic Process. Appl. 108 109–129.
• Chassagneux, J. F. (2012). An introduction to the numerical approximation of BSDEs. Lecture notes, 2nd Summer School of the Euro-Mediterranean Research Center for Mathematics and its Applications (EMRCMA). Available at www.imperial.ac.uk/~jchassag/.
• Chassagneux, J. F. (2013). Linear multi-step schemes for BSDEs. Preprint. Available at arXiv:1306.5548v1.
• Chassagneux, J. F. and Crisan, D. (2014). Runge–Kutta schemes for backward stochastic differential equations. Ann. Appl. Probab. 24 679–720.
• Chassagneux, J.-F. and Richou, A. (2013). Numerical simulation of quadratic BSDEs. Preprint. Available at arXiv:1307.5741.
• Crisan, D. and Manolarakis, K. (2010). Second order discretization of backward SDEs and simulation with the cubature method. Ann. Appl. Probab. 24 652–678.
• Crisan, D. and Manolarakis, K. (2012). Solving backward stochastic differential equations using the cubature method: Application to nonlinear pricing. SIAM J. Financial Math. 3 534–571.
• dos Reis, G., Réveillac, A. and Zhang, J. (2011). FBSDEs with time delayed generators: $L^{p}$-solutions, differentiability, representation formulas and path regularity. Stochastic Process. Appl. 121 2114–2150.
• El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
• Estep, D. J., Larson, M. G. and Williams, R. D. (2000). Estimating the error of numerical solutions of systems of reaction–diffusion equations. Mem. Amer. Math. Soc. 146 viii+109.
• 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.
• Gobet, E. and Turkedjiev, P. (2011). Approximation of discrete BSDE using least-squares regression. Technical Report hal-00642685.
• Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840. Springer, Berlin.
• Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2011). Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 1563–1576.
• Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2012). Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 1611–1641.
• Imkeller, P. (2008). Malliavin’s Calculus and Applications in Stochastic Control and Finance. IMPAN Lecture Notes 1. Polish Academy of Sciences, Institute of Mathematics, Warsaw.
• Imkeller, P. and dos Reis, G. (2010a). Path regularity and explicit convergence rate for BSDE with truncated quadratic growth. Stochastic Process. Appl. 120 348–379.
• Imkeller, P. and dos Reis, G. (2010b). Corrigendum to “Path regularity and explicit convergence rate for BSDE with truncated quadratic growth” [Stochastic Process. Appl. 120 (2010) 348–379] [MR2584898]. Stochastic Process. Appl. 120 2286–2288.
• Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
• Kovács, B. (2011). Semilinear parabolic problems. Master’s thesis, Eötvös Loránd Univ., Budapest.
• Lionnet, A. (2014). Topics on backward stochastic differential equations. Theoretical and practical aspects. Ph.D. thesis, Oxford Univ.
• Ma, J. and Zhang, J. (2002). Path regularity for solutions of backward stochastic differential equations. Probab. Theory Related Fields 122 163–190.
• Mao, X. and Szpruch, L. (2013). Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients. Stochastics 85 144–171.
• Matoussi, A. and Xu, M. (2008). Sobolev solution for semilinear PDE with obstacle under monotonicity condition. Electron. J. Probab. 13 1035–1067.
• Milstein, G. N. and Tretyakov, M. V. (2004). Stochastic Numerics for Mathematical Physics. Scientific Computation. Springer, Berlin.
• Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
• Pardoux, É. (1999). BSDEs, weak convergence and homogenization of semilinear PDEs. In Nonlinear Analysis, Differential Equations and Control (Montreal, QC, 1998). NATO Sci. Ser. C Math. Phys. Sci. 528 503–549. Kluwer Academic, Dordrecht.
• Rothe, F. (1984). Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Math. 1072. Springer, Berlin.
• Süli, E. and Mayers, D. F. (2003). An Introduction to Numerical Analysis. Cambridge Univ. Press, Cambridge.
• Touzi, N. (2013). Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs 29. Springer, New York.
• Zeidler, E. (1990). Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators. Springer, New York. Translated from the German by the author and Leo F. Boron.
• Zhang, G., Gunzburger, M. and Zhao, W. (2013). A sparse-grid method for multi-dimensional backward stochastic differential equations. J. Comput. Math. 31 221–248.
• Zhang, Q. and Zhao, H. (2012). Probabilistic representation of weak solutions of partial differential equations with polynomial growth coefficients. J. Theoret. Probab. 25 396–423.