Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Localization errors in solving stochastic partial differential equations in the whole space


Authors: Máté Gerencsér and István Gyöngy
Journal: Math. Comp. 86 (2017), 2373-2397
MSC (2010): Primary 60H15, 60H35, 65M06
DOI: https://doi.org/10.1090/mcom/3201
Published electronically: November 28, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius $ R$. This localization reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localization and the space and time discretization, and thus is fully implementable.


References [Enhancements On Off] (What's this?)

  • [1] Arnaud Debussche and Jacques Printems, Weak order for the discretization of the stochastic heat equation, Math. Comp. 78 (2009), no. 266, 845-863. MR 2476562, https://doi.org/10.1090/S0025-5718-08-02184-4
  • [2] Hongjie Dong and Nicolai V. Krylov, Rate of convergence of finite-difference approximations for degenerate linear parabolic equations with $ C^1$ and $ C^2$ coefficients, Electron. J. Differential Equations (2005), No. 102, 25. MR 2162263
  • [3] M. I. Freĭdlin, The factorization of nonnegative definite matrices, Teor. Verojatnost. i Primenen. 13 (1968), 375-378 (Russian, with English summary). MR 0229669
  • [4] László Gerencsér, On a class of mixing processes, Stochastics Stochastics Rep. 26 (1989), no. 3, 165-191. MR 1018543
  • [5] Máté Gerencsér and István Gyöngy, Finite difference schemes for stochastic partial differential equations in Sobolev spaces, Appl. Math. Optim. 72 (2015), no. 1, 77-100. MR 3369398, https://doi.org/10.1007/s00245-014-9272-2
  • [6] Máté Gerencsér, István Gyöngy, and Nicolai Krylov, On the solvability of degenerate stochastic partial differential equations in Sobolev spaces, Stoch. Partial Differ. Equ. Anal. Comput. 3 (2015), no. 1, 52-83. MR 3312592, https://doi.org/10.1007/s40072-014-0042-6
  • [7] István Gyöngy, On stochastic finite difference schemes, Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014), no. 4, 539-583. MR 3274891, https://doi.org/10.1007/s40072-014-0039-1
  • [8] István Gyöngy and Nicolai Krylov, On the rate of convergence of splitting-up approximations for SPDEs, Stochastic inequalities and applications, Progr. Probab., vol. 56, Birkhäuser, Basel, 2003, pp. 301-321. MR 2073438
  • [9] István Gyöngy and Annie Millet, On discretization schemes for stochastic evolution equations, Potential Anal. 23 (2005), no. 2, 99-134. MR 2139212, https://doi.org/10.1007/s11118-004-5393-6
  • [10] István Gyöngy and Nicolai Krylov, Accelerated finite difference schemes for linear stochastic partial differential equations in the whole space, SIAM J. Math. Anal. 42 (2010), no. 5, 2275-2296. MR 2729440, https://doi.org/10.1137/090781395
  • [11] Erika Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal. 18 (2003), no. 2, 141-186. MR 1953619, https://doi.org/10.1023/A:1020552804087
  • [12] Norbert Hilber, Oleg Reichmann, Christoph Schwab, and Christoph Winter, Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing, Springer Finance, Springer, Heidelberg, 2013. MR 3026658
  • [13] Arnulf Jentzen and Peter E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2102, 649-667. MR 2471778, https://doi.org/10.1098/rspa.2008.0325
  • [14] Peter E. Kloeden, Eckhard Platen, and Henri Schurz, Numerical solution of SDE through computer experiments, Universitext, Springer-Verlag, Berlin, 1994. With 1 IBM-PC floppy disk (3.5 inch; HD). MR 1260431
  • [15] N. V. Krylov, Controlled Diffusion Processes, Applications of Mathematics, vol. 14, Springer-Verlag, New York-Berlin, 1980. Translated from the Russian by A. B. Aries. MR 601776
  • [16] N. V. Krylov, On the Itô-Wentzell formula for distribution-valued processes and related topics, Probab. Theory Related Fields 150 (2011), no. 1-2, 295-319. MR 2800911, https://doi.org/10.1007/s00440-010-0275-x
  • [17] N. V. Krylov, A $ W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields 98 (1994), no. 3, 389-421. MR 1262972, https://doi.org/10.1007/BF01192260
  • [18] N. V. Krylov and B. L. Rozovskiĭ, Stochastic partial differential equations and diffusion processes, Uspekhi Mat. Nauk 37 (1982), no. 6(228), 75-95; English transl., Russian Math. Surveys 37 (1982), no. 6, 81-105 (Russian). MR 683274
  • [19] Hiroshi Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1997. Reprint of the 1990 original. MR 1472487
  • [20] Damien Lamberton and Bernard Lapeyre, Introduction to stochastic calculus applied to finance, Chapman & Hall, London, 1996. Translated from the 1991 French original by Nicolas Rabeau and François Mantion. MR 1422250
  • [21] James-Michael Leahy and Remigijus Mikulevičius, On classical solutions of linear stochastic integro-differential equations, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 3, 535-591. MR 3538010, https://doi.org/10.1007/s40072-016-0070-5
  • [22] L. F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, Philos. Trans. Roy. Soc. London Ser. A 210 (1910), 307-357.
  • [23] David Šiška, Error estimates for approximations of American put option price, Comput. Methods Appl. Math. 12 (2012), no. 1, 108-120. MR 3041004, https://doi.org/10.2478/cmam-2012-0007
  • [24] J. B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Anal. 23 (2005), no. 1, 1-43. MR 2136207, https://doi.org/10.1007/s11118-004-2950-y
  • [25] Yubin Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal. 43 (2005), no. 4, 1363-1384 (electronic). MR 2182132, https://doi.org/10.1137/040605278
  • [26] Hyek Yoo, Semi-discretization of stochastic partial differential equations on $ {\bf R}^1$ by a finite-difference method, Math. Comp. 69 (2000), no. 230, 653-666. MR 1654030, https://doi.org/10.1090/S0025-5718-99-01150-3

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 60H15, 60H35, 65M06

Retrieve articles in all journals with MSC (2010): 60H15, 60H35, 65M06


Additional Information

Máté Gerencsér
Affiliation: School of Mathematics and Maxwell Institute, The University of Edinburgh, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD Scotland, United Kingdom
Address at time of publication: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Email: mate.gerencser@ist.ac.at

István Gyöngy
Affiliation: School of Mathematics and Maxwell Institute, The University of Edinburgh, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD Scotland, United Kingdom
Email: gyongy@maths.ed.ac.uk

DOI: https://doi.org/10.1090/mcom/3201
Keywords: Cauchy problem, degenerate stochastic parabolic PDEs, localization error, finite difference method
Received by editor(s): August 22, 2015
Received by editor(s) in revised form: February 27, 2016
Published electronically: November 28, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society