## Bernoulli

• Bernoulli
• Volume 19, Number 1 (2013), 93-114.

### A numerical scheme for backward doubly stochastic differential equations

Auguste Aman

#### Abstract

In this paper we propose a numerical scheme for the class of backward doubly stochastic differential equations (BDSDEs) with possible path-dependent terminal values. We prove that our scheme converges in the strong $L^{2}$-sense and derives its rate of convergence. As an intermediate step we derive an $L^{2}$-type regularity of the solution to such BDSDEs. Such a notion of regularity, which can be thought of as the modulus of continuity of the paths in an $L^{2}$-sense, is new.

#### Article information

Source
Bernoulli, Volume 19, Number 1 (2013), 93-114.

Dates
First available in Project Euclid: 18 January 2013

https://projecteuclid.org/euclid.bj/1358531742

Digital Object Identifier
doi:10.3150/11-BEJ391

Mathematical Reviews number (MathSciNet)
MR3019487

Zentralblatt MATH identifier
1274.60217

#### Citation

Aman, Auguste. A numerical scheme for backward doubly stochastic differential equations. Bernoulli 19 (2013), no. 1, 93--114. doi:10.3150/11-BEJ391. https://projecteuclid.org/euclid.bj/1358531742

#### References

• [1] Aman, A. (2010). Reflected generalized backward doubly SDEs driven by Lévy processes and applications. J. Theoret. Probab. DOI:10.1007/s10959-010-0328-1.
• [2] Aman, A., N’zi, M. and Owo, J.M. (2010). A note on homeomorphism for backward doubly SDEs and applications. Stoch. Dyn. 10 1–12.
• [3] Bahlali, S. and Gherbal, B. (2010). Optimality conditions of controlled backward doubly stochastic differential equations. Random Oper. Stoch. Equ. 18 247–265.
• [4] Bally, V. (1997). Approximation scheme for solutions of BSDE. In Backward Stochastic Differential Equations (Paris, 19951996). Pitman Res. Notes Math. Ser. 364 177–191. Harlow: Longman.
• [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] Briand, P., Delyon, B. and Mémin, J. (2001). Donsker-type theorem for BSDEs. Electron. Commun. Probab. 6 1–14 (electronic).
• [7] Buckdahn, R. and Ma, J. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stochastic Process. Appl. 93 181–204.
• [8] Buckdahn, R. and Ma, J. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II. Stochastic Process. Appl. 93 205–228.
• [9] Chevance, D. (1997). Numerical methods for backward stochastic differential equations. In Numerical Methods in Finance (L.C.G. Rogers and D. Talay, eds.). Publ. Newton Inst. 232–244. Cambridge: Cambridge Univ. Press.
• [10] Douglas, J. Jr., Ma, J. and Protter, P. (1996). Numerical methods for forward–backward stochastic differential equations. Ann. Appl. Probab. 6 940–968.
• [11] Karatzas, I. and Shreve, S.E. (1998). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. New York: Springer.
• [12] Kloeden, P.E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Berlin: Springer.
• [13] Lemor, J.P., Gobet, E. and Warin, X. (2006). Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12 889–916.
• [14] Ma, J., Protter, P., San Martín, J. and Torres, S. (2002). Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12 302–316.
• [15] Ma, J., Protter, P. and Yong, J.M. (1994). Solving forward–backward stochastic differential equations explicitly—A four step scheme. Probab. Theory Related Fields 98 339–359.
• [16] Ma, J. and Zhang, J. (2002). Path regularity for solutions of backward SDE’s. Probab. Theory Related Fields 122 163–190.
• [17] Milstein, G.N. and Tretyakov, M.V. (2006). Numerical algorithms for forward–backward stochastic differential equations. SIAM J. Sci. Comput. 28 561–582 (electronic).
• [18] Pardoux, É. and Peng, S.G. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Related Fields 98 209–227.
• [19] Zhang, J. (2001). Some fine properties of backward stochastic differential equations. Ph.D. thesis, Purdue Univ.
• [20] Zhang, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459–488.