We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of dominant principal components. The focus of the present article is the derivation of sharp error bounds for the constant coefficient case and a first and second order approximation. We give a precise characterisation when these bounds hold for (non-smooth) option pricing applications and provide numerical results demonstrating that the practically observed convergence speed is in agreement with the theoretical predictions.
Accepté le :
DOI : 10.1051/m2an/2017003
Mots clés : High-dimensional PDEs, asymptotic expansions, anchored ANOVA, error bounds, financial derivative pricing
@article{M2AN_2017__51_6_2435_0,
author = {Reisinger, Christoph and Wissmann, Rasmus},
title = {Error analysis of truncated expansion solutions to high-dimensional parabolic {PDEs}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2435--2463},
publisher = {EDP-Sciences},
volume = {51},
number = {6},
year = {2017},
doi = {10.1051/m2an/2017003},
mrnumber = {3745177},
zbl = {1382.65280},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017003/}
}
TY - JOUR AU - Reisinger, Christoph AU - Wissmann, Rasmus TI - Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 2435 EP - 2463 VL - 51 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017003/ DO - 10.1051/m2an/2017003 LA - en ID - M2AN_2017__51_6_2435_0 ER -
%0 Journal Article %A Reisinger, Christoph %A Wissmann, Rasmus %T Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 2435-2463 %V 51 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017003/ %R 10.1051/m2an/2017003 %G en %F M2AN_2017__51_6_2435_0
Reisinger, Christoph; Wissmann, Rasmus. Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2435-2463. doi : 10.1051/m2an/2017003. http://www.numdam.org/articles/10.1051/m2an/2017003/
, A finite-difference method of higher-order accuracy for the solution of three-dimensional transient heat conduction problems. AIChE J. 7 (1961) 367–370. | DOI
and , Sparse grids. Acta Numer. 13 (2004) 1–123. | MR | Zbl
, and , Improved alternating-direction implicit method for solving transient three-dimensional heat diffusion problems. Numer. Heat Trans. Part B: Fundamentals 19 (1991) 69–84. | DOI
L.C. Evans, Partial Differential Equations. AMS, Providence, 1st edition (2000). | MR | Zbl
and . Dimension-adaptive tensor-product quadrature. Comput. 71 (2003) 65–87. | DOI | MR | Zbl
, and , A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen 36 (2013) 53–78. | DOI | MR | Zbl
and , Dimension-wise integration of high-dimensional functions with applications to finance. J. Complex. 26 (2010) 455–489. | DOI | MR | Zbl
and , Dimensionality reduction of high-dimensional data with a nonlinear principal component aligned generative topographic mapping. SIAM J. Sci. Comput. 36 (2014) 1027–1047. | DOI | MR | Zbl
, and , The smoothing effect of the ANOVA decomposition. J. Complex. 26 (2010) 523–551. | DOI | MR | Zbl
L.G. Gyurko and T. Lyons, Efficient and practical implementations of cubature on Wiener space. In Stoch. Anal.. Springer (2011) 73–111. | MR | Zbl
T. Haentjens and K.J. in ‘t Hout, ADI finite difference discretization of the Heston-Hull-White PDE. In Vol. 1281 of ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010. AIP Publishing (2010) 1995–1999.
, and , A highly parallel Black-Scholes solver based on adaptive sparse grids. Int. J. Comput. Math. 89 (2012) 1212–1238. | DOI | MR | Zbl
N. Hilber, S. Kehtari, C. Schwab and C. Winter, Wavelet finite element method for option pricing in highdimensional diffusion market models. Technical Report 2010–01, SAM, ETH Zürich (2010).
J. Imai and K.S. Tan, Minimizing effective dimension using linear transformation. In Monte Carlo and Quasi-Monte Carlo Methods 2002. Springer (2004). 275–292 | MR | Zbl
and , ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Model. 7 (2010) 303–320. | MR | Zbl
R. Sircar J.-P. Fouque, G. Papanicolaou and K. Sølna, Multiscale stochastic volatility for equity, interest rate, and credit derivatives. Cambridge University Press (2011). | MR | Zbl
, and , Low-rank tensor structure of linear diffusion operators in the TT and QTT formats. Lin. Algebra Appl. 438 (2013) 4204–4221. | DOI | MR | Zbl
K. Königsberger, Analysis 2. Springer (2004). | Zbl
, , and , On decompositions of multivariate functions. Math. Comput. 79 (2010) 953–966. | DOI | MR | Zbl
and , On coordinate transformation and grid stretching for sparse grid pricing of basket options. J. Comput. Appl. Math. 222 (2008) 193–209. | DOI | MR | Zbl
, Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103 (1992) 16–42. | DOI | MR | Zbl
and , High order recombination and an application to cubature on Wiener space. Ann. Appl. Prob. 22 (2012) 1301–1327. | DOI | MR | Zbl
and , Cubature on Wiener space. Proc. Royal Soc. A 460 (2004) 169–198. | DOI | MR | Zbl
and , Jr. The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3 (1955) 28–41. | DOI | MR | Zbl
, , and , Improved radial basis function methods for multi-dimensional option pricing. J. Comput. Appl. Math. 222 (2008) 82–93. | DOI | MR | Zbl
C. Reisinger, Asymptotic expansion around principal components and the complexity of dimension adaptive algorithms. In Sparse Grids and Applications, edited by J. Garcke and M. Griebel. Vol. 88 of Springer Lectures Notes in Computational Science and Engineering (2012) 263–276. | MR
and , Numerical valuation of derivatives in high-dimensional settings via PDE expansions. J. Comput. Fin. 18 (2015).
C. Reisinger and R. Wissmann, Finite difference methods for medium- and high-dimensional derivative pricing PDEs. In High-Performance Computing in Finance: Problems, Methods, and Solutions, edited by E. Vynckier, J. Kanniainen and J. Keane. Chapman and Hall/CRC, London (2017).
and , Efficient hierarchical approximation of high-dimensional option pricing problems. SIAM J. Sci. Comput. 29 (2007) 440–458. | DOI | MR | Zbl
P. Schröder, T. Gerstner and G. Wittum, Taylor-like ANOVA-expansion for high dimensional problems in finance. Working paper (2012).
P. Schröder, P. Mlynczak and G. Wittum, Dimension-wise decompositions and their efficient parallelization. In edited by T Gerstner and P.E. Kloeden. Recent Developments in Computational Finance: Foundations, Algorithms and Applications. Vol. 14 of Interdisciplinary Mathematical Sciences. World Scientific (2013). | MR | Zbl
, and , Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal. 40 (2002) 1650–1665. | DOI | MR | Zbl
, and , On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput. 71 (2002) 1609–1640. | DOI | MR | Zbl
and , Numerical solution of parabolic equations in high dimensions. ESAIM: M2AN 38 (2004) 93–127. | DOI | Numdam | MR | Zbl
, On the approximation error in high dimensional model representation. Edited by S.J. Mason, R.R. Hill, L. Mönch, O. Rose, T. Jefferson and J.W. Fowle. Winter Simulation Conference 2008 (2008) 453–462.
and , Why are high-dimension finance problems often of low effective dimension? SIAM J. Sci. Comput. 27 (2005) 159–183. | DOI | MR | Zbl
R. Wissmann, Expansion methods for high-dimensional PDEs in finance. DPhil thesis, University of Oxford (2015).
, and , Error estimates for the ANOVA method with polynomial chaos interpolation: tensor product functions. SIAM J. Sci. Comput. 34 (2012) 1165–1186. | DOI | MR | Zbl
and , Multi-factor financial derivatives on finite domains. Commun. Math. Sci. 1 (2003) 343–359. | DOI | MR | Zbl
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