Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs

Yu Fu; Weidong Zhao; Tao Zhou

doi:10.3934/dcdsb.2017174

Article Contents

2017, Volume 22, Issue 9: 3439-3458. Doi: 10.3934/dcdsb.2017174

This issue Previous Article The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations Next Article Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group

Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2.
School of Mathematics & Institute of Finance, Shandong University, Jinan 250100, China
3.
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author
* Corresponding author

Received: March 31, 2016

Revised: March 31, 2017

Published: October 2017

This work was supported by the National Natural Science Foundations of China under grants 91630312,91630203,11571351,11171189 and 11571206. The last author was also supported by NCMIS

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Abstract

This is the second part of a series papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high-dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high-dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discretization. The sparse grid Gaussian-Hermite quadrature rule is used to approximate the conditional expectations. And for the associated high-dimensional interpolations, we adopt a spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use the sparse grid approximations to recover the expansion coefficients. The FFT algorithm is used to speed up the recovery procedure, and the entire algorithm admits efficient and highly accurate approximations in high dimensions. Several numerical examples are presented to demonstrate the efficiency of the proposed methods.

Keywords:

Spectral method,
sparse grid approximations,
forward backward stochastic differential equations,
conditional expectations,
fast Fourier transform.

Mathematics Subject Classification: 60H35, 65C20, 60H10.

Citation:

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Figure 1. Sparse grid for CGL $(x_1, x_2)\in\mathcal{C}_2^6$ (Left) and sparse grid for GH $(x_1, x_2)\in\mathcal{G}_2^6$ (Right)

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Figure 2. Dimensions v.s. the running time for Example 3 with $N=128$

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Table 1. Main techniques used in the SSG and LTG methods for solving FBSDEs. TP: tensor product. SG: sparse grid. GH: Gaussian-Hermite

Method	Meshes	Conditional expectations	Approximation & interpolation
SSG	sparse grid	SG GH quadrature	SG interpolation
LTG	TP uniform mesh	TP GH quadrature	Lagrangian

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Table 2. Errors and convergence rates for Example 1 by SSG (Top) and LTG (Bottom)

step number	errors	N=8	N=16	N=32	N=64	N=128	CR
1-step	$\mathcal{E}_Y$	3.991E-02	2.050E-02	1.039E-02	5.232E-03	2.625E-03	0.982
	$\mathcal{E}_Z$	5.186E-02	2.649E-02	1.339E-02	6.733E-03	3.377E-03	0.986
	RT	2.519	4.638	9.521	19.199	38.679
2-step	$\mathcal{E}_Y$	5.620E-03	1.456E-03	3.670E-04	9.182E-05	2.286E-05	1.987
	$\mathcal{E}_Z$	6.978E-03	1.847E-03	4.813E-04	1.225E-04	3.090E-05	1.955
	RT	6.852	15.596	33.401	68.541	143.552
3-step	$\mathcal{E}_Y$	9.748E-04	1.342E-04	1.728E-05	2.264E-06	8.196E-07	2.632
	$\mathcal{E}_Z$	3.091E-03	3.850E-04	4.757E-05	6.009E-06	8.834E-07	2.955
	RT	6.899	19.403	43.835	95.902	197.608
step number	errors	N=8	N=16	N=32	N=64	N=128	CR
1-step	$\mathcal{E}_Y$	7.204E-01	3.989E-01	1.849E-01	7.321E-02	2.871E-02	1.174
	$\mathcal{E}_Z$	2.670E-01	1.534E-01	7.409E-02	3.152E-02	1.332E-02	1.093
	RT	1.047	4.609	24.120	141.209	844.974
2-step	$\mathcal{E}_Y$	4.873E-01	1.105E-01	2.220E-02	4.165E-03	7.708E-04	2.333
	$\mathcal{E}_Z$	1.999E-01	4.706E-02	1.174E-02	2.494E-03	4.159E-04	2.205
	RT	3.861	12.970	60.661	363.052	2139.540
3-step	$\mathcal{E}_Y$	2.656E-01	2.252E-02	2.295E-03	2.247E-04	2.024E-05	3.401
	$\mathcal{E}_Z$	1.136E-01	9.841E-03	1.306E-03	1.417E-04	1.244E-05	3.243
	RT	13.010	43.490	193.784	968.788	4929.977

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Table 3. Errors and convergence rates for Example 2

step number	errors	N=8	N=16	N=32	N=64	N=128	CR
1-step	$\mathcal{E}_Y$	6.229E-03	3.173E-03	1.547E-03	7.695E-04	3.849E-04	1.008
	$\mathcal{E}_Z$	8.693E-02	4.335E-02	2.165E-02	1.082E-02	5.411E-03	1.001
	RT	4.268	9.036	17.720	35.740	76.049
2-step	$\mathcal{E}_Y$	4.126E-05	1.148E-05	2.251E-06	3.986E-07	1.051E-07	2.208
	$\mathcal{E}_Z$	5.470E-04	1.630E-04	4.222E-05	8.722E-06	2.061E-06	2.032
	RT	6.986	15.926	34.530	72.690	153.092
3-step	$\mathcal{E}_Y$	4.078E-05	8.408E-06	1.107E-06	1.370E-07	1.837E-08	2.817
	$\mathcal{E}_Z$	4.438E-04	5.143E-05	7.673E-06	1.049E-06	1.514E-07	2.865
	RT	7.980	20.650	48.610	102.595	212.189

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Table 4. Errors and convergence rates for Example 3

scheme	sparse grid		N=8	N=16	N=32	N=64	N=128	CR
1-step	$q=3$ $\mathcal{C}_3^4$ & $\mathcal{G}_3^4$	$\mathcal{E}_Y$	5.717E-02	2.999E-02	1.557E-02	8.021E-03	4.104E-03	0.950
		$\mathcal{E}_Z$	3.129E-02	1.575E-02	7.988E-03	4.057E-03	2.057E-03	0.981
		RT	0.107	0.226	0.389	0.627	1.268
2-step	$q=3$ $\mathcal{C}_3^4$ & $\mathcal{G}_3^4$	$\mathcal{E}_Y$	2.766E-03	7.861E-04	2.091E-04	5.396E-05	1.371E-05	1.918
		$\mathcal{E}_Z$	4.833E-03	1.197E-03	3.006E-04	7.588E-05	1.916E-05	1.994
		RT	0.131	0.235	0.509	1.067	2.187
3-step	$q=3$ $\mathcal{C}_3^4$ & $\mathcal{G}_3^4$	$\mathcal{E}_Y$	1.244E-04	1.183E-05	1.061E-06	9.522E-08	8.587E-09	3.460
		$\mathcal{E}_Z$	3.425E-04	4.375E-05	5.575E-06	7.067E-07	8.922E-08	2.976
		RT	0.173	0.370	0.750	1.478	3.072
1-step	$q=4$ $\mathcal{C}_4^5$ & $\mathcal{G}_4^5$	$\mathcal{E}_Y$	1.196E-01	6.208E-02	3.187E-02	1.626E-02	8.259E-03	0.965
		$\mathcal{E}_Z$	3.445E-02	1.735E-02	8.772E-03	4.439E-03	2.244E-03	0.985
		RT	1.682	3.153	6.178	12.934	26.182
2-step	$q=4$ $\mathcal{C}_4^5$ & $\mathcal{G}_4^5$	$\mathcal{E}_Y$	9.114E-03	2.817E-03	7.851E-04	2.087E-04	5.426E-05	1.854
		$\mathcal{E}_Z$	6.769E-03	1.628E-03	4.038E-04	1.012E-04	2.547E-05	2.012
		RT	2.513	4.442	9.606	20.160	41.504
3-step	$q=4$ $\mathcal{C}_4^5$ & $\mathcal{G}_4^5$	$\mathcal{E}_Y$	4.879E-04	7.611E-05	1.052E-05	1.378E-06	1.763E-07	2.865
		$\mathcal{E}_Z$	1.176E-03	1.420E-04	1.767E-05	2.215E-06	2.786E-07	3.009
		RT	1.993	5.722	13.296	28.614	59.671
1-step	$q=5$ $\mathcal{C}_5^6$ & $\mathcal{G}_5^6$	$\mathcal{E}_Y$	2.269E-01	1.177E-01	6.023E-02	3.061E-02	1.549E-02	0.969
		$\mathcal{E}_Z$	4.345E-02	2.196E-02	1.110E-02	5.608E-03	2.829E-03	0.985
		RT	23.196	47.566	99.318	203.587	411.991
2-step	$q=5$ $\mathcal{C}_5^6$ & $\mathcal{G}_5^6$	$\mathcal{E}_Y$	2.941E-02	8.713E-03	2.376E-03	6.223E-04	1.600E-04	1.885
		$\mathcal{E}_Z$	9.182E-03	2.216E-03	5.534E-04	1.393E-04	3.510E-05	2.005
		RT	33.610	80.876	177.315	372.765	766.094
3-step	$q=5$ $\mathcal{C}_5^6$ & $\mathcal{G}_5^6$	$\mathcal{E}_Y$	2.549E-03	4.157E-04	6.185E-05	8.531E-06	1.127E-06	2.789
		$\mathcal{E}_Z$	2.364E-03	2.687E-04	3.254E-05	4.031E-06	5.037E-07	3.045
		RT	36.049	105.507	245.986	530.853	1106.870
1-step	$q=6$ $\mathcal{C}_6^7$ & $\mathcal{G}_6^7$	$\mathcal{E}_Y$	4.068E-01	2.117E-01	1.084E-01	5.500E-02	2.779E-02	0.969
		$\mathcal{E}_Z$	5.899E-02	2.996E-02	1.516E-02	7.652E-03	3.855E-03	0.984
		RT	368.274	792.758	1640.774	3346.529	6758.066
2-step	$q=6$ $\mathcal{C}_6^7$ & $\mathcal{G}_6^7$	$\mathcal{E}_Y$	7.236E-02	2.130E-02	5.783E-03	1.510E-03	3.869E-04	1.891
		$\mathcal{E}_Z$	1.307E-02	3.277E-03	8.382E-04	2.134E-04	5.405E-05	1.978
		RT	1112.524	2813.641	6114.390	12814.473	26454.702
3-step	$q=6$ $\mathcal{C}_6^7$ & $\mathcal{G}_6^7$	$\mathcal{E}_Y$	1.037E-02	1.742E-03	2.528E-04	3.407E-05	4.433E-06	2.806
		$\mathcal{E}_Z$	3.924E-03	4.391E-04	5.339E-05	6.647E-06	8.328E-07	3.045
		RT	594.060	1767.938	4110.876	8853.131	18481.311

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Table 5. Errors and convergence rates for Example 4

scheme	sparse gird		N=8	N=16	N=32	N=64	N=128	CR
1-step	$q=2$ $\mathcal{C}_2^3$ & $\mathcal{G}_2^3$	$\mathcal{E}_Y$	4.246E-03	1.925E-03	9.181E-04	4.508E-04	2.243E-04	1.058
		$\mathcal{E}_Z$	1.149E-02	4.898E-03	2.098E-03	9.057E-04	3.948E-04	1.216
		RT	0.030	0.045	0.083	0.160	0.267
2-step	$q=2$ $\mathcal{C}_2^3$ & $\mathcal{G}_2^3$	$\mathcal{E}_Y$	4.093E-04	8.289E-05	1.646E-05	3.207E-06	6.101E-07	2.347
		$\mathcal{E}_Z$	5.429E-03	1.395E-03	3.568E-04	9.099E-05	2.315E-05	1.968
		RT	0.032	0.059	0.090	0.171	0.327
3-step	$q=2$ $\mathcal{C}_2^3$ & $\mathcal{G}_2^3$	$\mathcal{E}_Y$	2.459E-05	2.651E-06	2.695E-07	2.658E-08	2.538E-09	3.312
		$\mathcal{E}_Z$	5.266E-05	5.713E-06	6.099E-07	6.276E-08	6.215E-09	3.261
		RT	0.033	0.075	0.132	0.276	0.430
1-step	$q=3$ $\mathcal{C}_3^4$ & $\mathcal{G}_3^4$	$\mathcal{E}_Y$	2.463E-03	1.068E-03	4.812E-04	2.229E-04	1.072E-04	1.130
		$\mathcal{E}_Z$	3.243E-03	1.363E-03	5.990E-04	2.630E-04	1.154E-04	1.200
		RT	0.295	0.422	0.762	1.420	2.690
2-step	$q=3$ $\mathcal{C}_3^4$ & $\mathcal{G}_3^4$	$\mathcal{E}_Y$	2.044E-04	7.910E-05	1.676E-05	3.397E-06	6.738E-07	2.103
		$\mathcal{E}_Z$	1.073E-03	2.444E-04	6.285E-05	1.632E-05	4.227E-06	1.988
		RT	0.352	0.717	1.385	2.610	4.946
3-step	$q=3$ $\mathcal{C}_3^4$ & $\mathcal{G}_3^4$	$\mathcal{E}_Y$	8.103E-06	1.743E-06	1.890E-07	1.942E-08	1.933E-09	3.055
		$\mathcal{E}_Z$	7.540E-06	1.577E-06	1.795E-07	1.933E-08	1.995E-09	3.012
		RT	0.364	1.223	1.936	3.743	7.150
1-step	$q=4$ $\mathcal{C}_4^5$ & $\mathcal{G}_4^5$	$\mathcal{E}_Y$	1.663E-03	7.351E-04	3.068E-04	1.370E-04	6.294E-05	1.187
		$\mathcal{E}_Z$	1.360E-03	5.763E-04	2.374E-04	1.068E-04	4.838E-05	1.206
		RT	3.916	8.050	15.738	29.987	55.418
2-step	$q=4$ $\mathcal{C}_4^5$ & $\mathcal{G}_4^5$	$\mathcal{E}_Y$	6.776E-05	4.165E-05	1.569E-05	3.408E-06	6.925E-07	1.684
		$\mathcal{E}_Z$	3.665E-04	8.033E-05	1.727E-05	4.479E-06	1.191E-06	2.070
		RT	5.064	13.279	28.974	56.345	105.155
3-step	$q=4$ $\mathcal{C}_4^5$ & $\mathcal{G}_4^5$	$\mathcal{E}_Y$	4.932E-06	1.065E-06	1.111E-07	1.115E-08	1.087E-09	3.087
		$\mathcal{E}_Z$	2.474E-06	5.495E-07	6.062E-08	6.348E-09	6.390E-10	3.027
		RT	34.015	117.047	250.853	483.481	926.454
1-step	$q=5$ $\mathcal{C}_5^6$ & $\mathcal{G}_5^6$	$\mathcal{E}_Y$	1.167E-03	5.489E-04	2.341E-04	9.343E-05	4.184E-05	1.216
		$\mathcal{E}_Z$	6.751E-04	3.055E-04	1.268E-04	5.183E-05	2.408E-05	1.218
		RT	58.937	134.345	289.234	570.587	1049.247
2-step	$q=5$ $\mathcal{C}_5^6$ & $\mathcal{G}_5^6$	$\mathcal{E}_Y$	6.572E-05	4.173E-05	1.246E-05	2.548E-06	5.087E-07	1.806
		$\mathcal{E}_Z$	1.464E-04	2.885E-05	6.466E-06	1.763E-06	4.792E-07	2.054
		RT	736.405	2004.185	4124.286	7776.645	14561.700
3-step	$q=5$ $\mathcal{C}_5^6$ & $\mathcal{G}_5^6$	$\mathcal{E}_Y$	1.649E-06	6.334E-07	9.009E-08	9.209E-09	9.147E-10	2.774
		$\mathcal{E}_Z$	5.318E-07	2.072E-07	3.079E-08	3.274E-09	3.372E-10	2.723
		RT	709.587	2599.783	5916.660	11366.633	21380.402

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