Cubature method to solve BSDEs: Error expansion and complexity control

Chassagneux, Jean-Francois; Garcia Trillos, Camilo

doi:10.1090/mcom/3522

Cubature method to solve BSDEs: Error expansion and complexity control
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by Jean-Francois Chassagneux and Camilo A. Garcia Trillos HTML | PDF

Math. Comp. 89 (2020), 1895-1932 Request permission

Abstract:

We obtain an explicit error expansion for the solution of Backward Stochastic Differential Equations (BSDEs) using the cubature on Wiener spaces method. The result is proved under a mild strengthening of the assumptions needed for the application of the cubature method. The explicit expansion can then be used to construct implementable higher order approximations via Richardson-Romberg extrapolation. To allow for an effective efficiency improvement of the interpolated algorithm, we introduce an additional projection on finite grids through interpolation operators. We study the resulting complexity reduction in the case of the linear interpolation.

References

Vlad Bally and Gilles Pagès, A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems, Bernoulli 9 (2003), no. 6, 1003–1049. MR 2046816, DOI 10.3150/bj/1072215199
Christian Bender and Jianfeng Zhang, Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab. 18 (2008), no. 1, 143–177. MR 2380895, DOI 10.1214/07-AAP448
Bruno Bouchard and Nizar Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl. 111 (2004), no. 2, 175–206. MR 2056536, DOI 10.1016/j.spa.2004.01.001
Philippe Briand, Bernard Delyon, and Jean Mémin, Donsker-type theorem for BSDEs, Electron. Comm. Probab. 6 (2001), 1–14. MR 1817885, DOI 10.1214/ECP.v6-1030
Philippe Briand and Céline Labart, Simulation of BSDEs by Wiener chaos expansion, Ann. Appl. Probab. 24 (2014), no. 3, 1129–1171. MR 3199982, DOI 10.1214/13-AAP943
Hans-Joachim Bungartz and Michael Griebel, Sparse grids, Acta Numer. 13 (2004), 147–269. MR 2249147, DOI 10.1017/S0962492904000182
Jean-François Chassagneux and Adrien Richou, Numerical stability analysis of the Euler scheme for BSDEs, SIAM J. Numer. Anal. 53 (2015), no. 2, 1172–1193. MR 3338675, DOI 10.1137/140977047
Jean-François Chassagneux, Linear multistep schemes for BSDEs, SIAM J. Numer. Anal. 52 (2014), no. 6, 2815–2836. MR 3284573, DOI 10.1137/120902951
Jean-François Chassagneux and Adrien Richou, Rate of convergence for the discrete-time approximation of reflected BSDEs arising in switching problems, Stochastic Process. Appl. 129 (2019), no. 11, 4597–4637. MR 4013874, DOI 10.1016/j.spa.2018.12.009
Jean-François Chassagneux and Dan Crisan, Runge-Kutta schemes for backward stochastic differential equations, Ann. Appl. Probab. 24 (2014), no. 2, 679–720. MR 3178495, DOI 10.1214/13-AAP933
Jean-François Chassagneux, Dan Crisan, and François Delarue, Numerical method for FBSDEs of McKean-Vlasov type, Ann. Appl. Probab. 29 (2019), no. 3, 1640–1684. MR 3914553, DOI 10.1214/18-AAP1429
Jean-François Chassagneux and Adrien Richou, Numerical simulation of quadratic BSDEs, Ann. Appl. Probab. 26 (2016), no. 1, 262–304. MR 3449318, DOI 10.1214/14-AAP1090
P. E. Chaudru de Raynal and C. A. Garcia Trillos, A cubature based algorithm to solve decoupled McKean-Vlasov forward-backward stochastic differential equations, Stochastic Process. Appl. 125 (2015), no. 6, 2206–2255. MR 3322862, DOI 10.1016/j.spa.2014.11.018
Dan Crisan and François Delarue, Sharp derivative bounds for solutions of degenerate semi-linear partial differential equations, J. Funct. Anal. 263 (2012), no. 10, 3024–3101. MR 2973334, DOI 10.1016/j.jfa.2012.07.015
Dan Crisan and Terry Lyons, Minimal entropy approximations and optimal algorithms, Monte Carlo Methods Appl. 8 (2002), no. 4, 343–355. MR 1943203, DOI 10.1515/mcma.2002.8.4.343
D. Crisan and K. Manolarakis, Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing, SIAM J. Financial Math. 3 (2012), no. 1, 534–571. MR 2968045, DOI 10.1137/090765766
Dan Crisan and Konstantinos Manolarakis, Second order discretization of backward SDEs and simulation with the cubature method, Ann. Appl. Probab. 24 (2014), no. 2, 652–678. MR 3178494, DOI 10.1214/13-AAP932
D. Crisan, K. Manolarakis, and C. Nee, Cubature methods and applications, Paris-Princeton Lectures on Mathematical Finance 2013, Lecture Notes in Math., vol. 2081, Springer, Cham, 2013, pp. 203–316. MR 3183925, DOI 10.1007/978-3-319-00413-6_{4}
D. Crisan, K. Manolarakis, and N. Touzi, On the Monte Carlo simulation of BSDEs: an improvement on the Malliavin weights, Stochastic Process. Appl. 120 (2010), no. 7, 1133–1158. MR 2639741, DOI 10.1016/j.spa.2010.03.015
François Delarue and Stéphane Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab. 16 (2006), no. 1, 140–184. MR 2209339, DOI 10.1214/105051605000000674
Emmanuel Gobet and Céline Labart, Error expansion for the discretization of backward stochastic differential equations, Stochastic Process. Appl. 117 (2007), no. 7, 803–829. MR 2330720, DOI 10.1016/j.spa.2006.10.007
Emmanuel Gobet, Jean-Philippe Lemor, and Xavier Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab. 15 (2005), no. 3, 2172–2202. MR 2152657, DOI 10.1214/105051605000000412
Lajos Gergely Gyurkó and Terry J. Lyons, Efficient and practical implementations of cubature on Wiener space, Stochastic analysis 2010, Springer, Heidelberg, 2011, pp. 73–111. MR 2789080, DOI 10.1007/978-3-642-15358-7_{5}
Pierre Henry-Labordère, Xiaolu Tan, and Nizar Touzi, A numerical algorithm for a class of BSDEs via the branching process, Stochastic Process. Appl. 124 (2014), no. 2, 1112–1140. MR 3138609, DOI 10.1016/j.spa.2013.10.005
Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374, DOI 10.1007/978-3-662-12616-5
C. Litterer and T. Lyons, High order recombination and an application to cubature on Wiener space, Ann. Appl. Probab. 22 (2012), no. 4, 1301–1327. MR 2985162, DOI 10.1214/11-AAP786
Terry Lyons and Nicolas Victoir, Cubature on Wiener space, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2041, 169–198. Stochastic analysis with applications to mathematical finance. MR 2052260, DOI 10.1098/rspa.2003.1239
Colm Nee, Sharp gradient bounds for the diffusion semigroup, Imperial College London (University of London), 2011.
Gilles Pagès and Abass Sagna, Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering, Stochastic Process. Appl. 128 (2018), no. 3, 847–883. MR 3758340, DOI 10.1016/j.spa.2017.05.009
É. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Stochastic partial differential equations and their applications (Charlotte, NC, 1991) Lect. Notes Control Inf. Sci., vol. 176, Springer, Berlin, 1992, pp. 200–217. MR 1176785, DOI 10.1007/BFb0007334
Denis Talay and Luciano Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl. 8 (1990), no. 4, 483–509 (1991). MR 1091544, DOI 10.1080/07362999008809220
Guannan Zhang, Max Gunzburger, and Weidong Zhao, A sparse-grid method for multi-dimensional backward stochastic differential equations, J. Comput. Math. 31 (2013), no. 3, 221–248. MR 3063734, DOI 10.4208/jcm.1212-m4014
Jianfeng Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab. 14 (2004), no. 1, 459–488. MR 2023027, DOI 10.1214/aoap/1075828058