## The Annals of Applied Probability

### Runge–Kutta schemes for backward stochastic differential equations

#### Abstract

We study the convergence of a class of Runge–Kutta type schemes for backward stochastic differential equations (BSDEs) in a Markovian framework. The schemes belonging to the class under consideration benefit from a certain stability property. As a consequence, the overall rate of the convergence of these schemes is controlled by their local truncation error. The schemes are categorized by the number of intermediate stages implemented between consecutive partition time instances. We show that the order of the schemes matches the number $p$ of intermediate stages for $p\le3$. Moreover, we show that the so-called order barrier occurs at $p=3$, that is, that it is not possible to construct schemes of order $p$ with $p$ stages, when $p>3$. The analysis is done under sufficient regularity on the final condition and on the coefficients of the BSDE.

#### Article information

Source
Ann. Appl. Probab., Volume 24, Number 2 (2014), 679-720.

Dates
First available in Project Euclid: 10 March 2014

https://projecteuclid.org/euclid.aoap/1394465369

Digital Object Identifier
doi:10.1214/13-AAP933

Mathematical Reviews number (MathSciNet)
MR3178495

Zentralblatt MATH identifier
1303.60045

#### Citation

Chassagneux, Jean-François; Crisan, Dan. Runge–Kutta schemes for backward stochastic differential equations. Ann. Appl. Probab. 24 (2014), no. 2, 679--720. doi:10.1214/13-AAP933. https://projecteuclid.org/euclid.aoap/1394465369

#### References

• [1] Bouchard, B. and Elie, R. (2008). Discrete-time approximation of decoupled forward-backward SDE with jumps. Stochastic Process. Appl. 118 53–75.
• [2] Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
• [3] Butcher, J. C. (2008). Numerical Methods for Ordinary Differential Equations, 2nd ed. Wiley, Chichester.
• [4] 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.
• [5] Geiss, C., Geiss, S. and Gobet, E. (2012). Generalized fractional smoothness and $L_{p}$-variation of BSDEs with non-Lipschitz terminal condition. Stochastic Process. Appl. 122 2078–2116.
• [6] Gobet, E. and Labart, C. (2007). Error expansion for the discretization of backward stochastic differential equations. Stochastic Process. Appl. 117 803–829.
• [7] 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.
• [8] Gobet, E. and Makhlouf, A. (2010). $L_{2}$-time regularity of BSDEs with irregular terminal functions. Stochastic Process. Appl. 120 1105–1132.
• [9] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
• [10] Kusuoka, S. (2001). Approximation of expectation of diffusion process and mathematical finance. In Taniguchi Conference on Mathematics Nara ’98. Adv. Stud. Pure Math. 31 147–165. Math. Soc. Japan, Tokyo.
• [11] Lyons, T. and Victoir, N. (2004). Cubature on Wiener space. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 169–198.
• [12] Ninomiya, S. (2003). A new simulation scheme of diffusion processes: Application of the Kusuoka approximation to finance problems. Math. Comput. Simulation 62 479–486. 3rd IMACS Seminar on Monte Carlo Methods—MCM 2001 (Salzburg).
• [13] Ninomiya, S. (2003). A partial sampling method applied to the Kusuoka approximation. Monte Carlo Methods Appl. 9 27–38.
• [14] Ninomiya, S. and Victoir, N. (2008). Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15 107–121.
• [15] Pardoux, É. and Peng, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991). Lecture Notes in Control and Inform. Sci. 176 200–217. Springer, Berlin.
• [16] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
• [17] Zhang, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459–488.