Annals of Probability

Taylor expansions of solutions of stochastic partial differential equations with additive noise

Arnulf Jentzen and Peter Kloeden

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Abstract

The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an Itô formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of a SODE using an iterated application of the Itô formula. Consequently, until recently, only low order numerical approximation results for such a SPDE have been available. Here, the fact that the solution of a SPDE driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the SPDE provides an alternative approach for deriving stochastic Taylor expansions for the solution of such a SPDE. Essentially, the exponential factor has a mollifying effect and ensures that all integrals take values in the Hilbert space under consideration. The iteration of such integrals allows us to derive stochastic Taylor expansions of arbitrarily high order, which are robust in the sense that they also hold for other types of driving noise processes such as fractional Brownian motion. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.

Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 532-569.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1268143526

Digital Object Identifier
doi:10.1214/09-AOP500

Mathematical Reviews number (MathSciNet)
MR2642885

Zentralblatt MATH identifier
1220.35202

Subjects
Primary: 35K90: Abstract parabolic equations 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) 65C30: Stochastic differential and integral equations 65M99: None of the above, but in this section

Keywords
Taylor expansions stochastic partial differential equations SPDE strong convergence stochastic trees

Citation

Jentzen, Arnulf; Kloeden, Peter. Taylor expansions of solutions of stochastic partial differential equations with additive noise. Ann. Probab. 38 (2010), no. 2, 532--569. doi:10.1214/09-AOP500. https://projecteuclid.org/euclid.aop/1268143526


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References

  • [1] Burrage, K. and Burrage, P. M. (2000). Order conditions of stochastic Runge–Kutta methods by B-series. SIAM J. Numer. Anal. 38 1626–1646.
    Zentralblatt MATH: 0983.65006
    Digital Object Identifier: doi:10.1137/S0036142999363206
  • [2] Butcher, J. C. (1987). The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear methods. Wiley, Chichester.
  • [3] Berland, H., Owren, B. and Skaflestad, B. (2005). B-series and order conditions for exponential integrators. SIAM J. Numer. Anal. 43 1715–1727.
    Zentralblatt MATH: 1109.65060
    Digital Object Identifier: doi:10.1137/040612683
  • [4] Berland, H., Skaflestad, B. and Wright, W. (2007). Expint—A Matlab package for exponential integrators. ACM Trans. Math. Software 33 1–17.
  • [5] Cox, S. M. and Matthews, P. C. (2002). Exponential time differencing for stiff systems. J. Comput. Phys. 176 430–455.
    Zentralblatt MATH: 1005.65069
    Digital Object Identifier: doi:10.1006/jcph.2002.6995
  • [6] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
    Zentralblatt MATH: 0761.60052
  • [7] Davie, A. M. and Gaines, J. G. (2001). Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. Math. Comp. 70 121–134.
    Zentralblatt MATH: 0956.60064
    Digital Object Identifier: doi:10.1090/S0025-5718-00-01224-2
  • [8] Deuflhard, P. and Bornemann, F. (2002). Scientific Computing with Ordinary Differential Equations. Texts in Applied Mathematics 42. Springer, New York.
    Zentralblatt MATH: 1001.65071
  • [9] Engel, K. and Nagel, R. (1999). One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin.
    Zentralblatt MATH: 0952.47036
  • [10] Gradinaru, M., Nourdin, I. and Tindel, S. (2005). Itô’s- and Tanaka’s-type formulae for the stochastic heat equation: The linear case. J. Funct. Anal. 228 114–143.
    Mathematical Reviews (MathSciNet): MR2170986
    Zentralblatt MATH: 1084.60039
    Digital Object Identifier: doi:10.1016/j.jfa.2005.02.008
  • [11] Gyöngy, I. (1998). A note on Euler’s approximations. Potential Anal. 8 205–216.
  • [12] Gyöngy, I. (1998). Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I. Potential Anal. 9 1–25.
  • [13] Gyöngy, I. (1999). Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II. Potential Anal. 11 1–37.
  • [14] Gyöngy, I. and Krylov, N. (2003). On the splitting-up method and stochastic partial differential equations. Ann. Probab. 31 564–591.
    Mathematical Reviews (MathSciNet): MR1964941
    Zentralblatt MATH: 1028.60058
    Digital Object Identifier: doi:10.1214/aop/1048516528
    Project Euclid: euclid.aop/1048516528
  • [15] Hausenblas, E. (2003). Approximation for semilinear stochastic evolution equations. Potential Anal. 18 141–186.
    Zentralblatt MATH: 1015.60053
    Digital Object Identifier: doi:10.1023/A:1020552804087
  • [16] Hochbruck, M., Lubich, C. and Selhofer, H. (1998). Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19 1552–1574.
    Zentralblatt MATH: 0912.65058
    Digital Object Identifier: doi:10.1137/S1064827595295337
  • [17] Hochbruck, M. and Ostermann, A. (2005). Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43 1069–1090.
    Zentralblatt MATH: 1093.65052
    Digital Object Identifier: doi:10.1137/040611434
  • [18] Hochbruck, M. and Ostermann, A. (2005). Exponential Runge–Kutta methods for parabolic problems. Appl. Numer. Math. 53 323–339.
    Zentralblatt MATH: 1070.65099
    Digital Object Identifier: doi:10.1016/j.apnum.2004.08.005
  • [19] Hochbruck, M., Ostermann, A. and Schweitzer, J. (2008/09). Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47 786–803.
    Zentralblatt MATH: 1193.65119
    Digital Object Identifier: doi:10.1137/080717717
  • [20] Jentzen, A. (2007). Numerische Verfahren hoher Ordnung für zufällige Differentialgleichungen, Diplomarbeit. Johann Wolfgang Geothe Univ. Frankfurt am Main.
  • [21] Jentzen, A. (2009). Higher order pathwise numerical approximations of SPDEs with additive noise. Submitted.
    Zentralblatt MATH: 1176.60051
    Digital Object Identifier: doi:10.1007/s11118-009-9139-3
  • [22] Jentzen, A. (2009). Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients. Potential Anal. 4 375–404.
    Zentralblatt MATH: 1176.60051
    Digital Object Identifier: doi:10.1007/s11118-009-9139-3
  • [23] Jentzen, A. and Kloeden, P. E. (2009). Pathwise Taylor schemes for random ordinary differential equations. BIT 49 113–140.
    Zentralblatt MATH: 1162.65305
    Digital Object Identifier: doi:10.1007/s10543-009-0211-6
  • [24] Jentzen, A. and Kloeden, P. E. (2009). 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. 465 649–667.
    Zentralblatt MATH: 1186.65011
    Digital Object Identifier: doi:10.1098/rspa.2008.0325
  • [25] Jentzen, A. and Kloeden, P. E. (2009). The numerical approximation of stochastic partial differential equations. Milan J. Math. 77 205–244.
    Zentralblatt MATH: 1205.60130
    Digital Object Identifier: doi:10.1007/s00032-009-0100-0
  • [26] Jentzen, A., Kloeden, P. E. and Neuenkirch, A. (2009). Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. Numer. Math. 112 41–64.
    Zentralblatt MATH: 1163.65003
    Digital Object Identifier: doi:10.1007/s00211-008-0200-8
  • [27] Kassam, A.-K. and Trefethen, L. N. (2005). Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26 1214–1233.
    Zentralblatt MATH: 1077.65105
    Digital Object Identifier: doi:10.1137/S1064827502410633
  • [28] Kloeden, P. E. and Jentzen, A. (2007). Pathwise convergent higher order numerical schemes for random ordinary differential equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 2929–2944.
    Zentralblatt MATH: 1140.65015
    Digital Object Identifier: doi:10.1098/rspa.2007.0055
  • [29] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1214374
    Zentralblatt MATH: 0752.60043
  • [30] Krogstad, S. (2005). Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203 72–88.
    Zentralblatt MATH: 1063.65097
    Digital Object Identifier: doi:10.1016/j.jcp.2004.08.006
  • [31] Lawson, J. D. (1967). Generalized Runge–Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4 372–380.
    Zentralblatt MATH: 0223.65030
    Digital Object Identifier: doi:10.1137/0704033
  • [32] Müller-Gronbach, T. and Ritter, K. (2007). Lower bounds and nonuniform time discretization for approximation of stochastic heat equations. Found. Comput. Math. 7 135–181.
    Zentralblatt MATH: 1136.60044
    Digital Object Identifier: doi:10.1007/s10208-005-0166-6
  • [33] Müller-Gronbach, T. and Ritter, K. (2007). An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise. BIT 47 393–418.
    Zentralblatt MATH: 1127.65006
    Digital Object Identifier: doi:10.1007/s10543-007-0129-9
  • [34] Müller-Gronbach, T., Ritter, K. and Wagner, T. (2008). Optimal pointwise approximation of a linear stochastic heat equation with additive space-time white noise. In Monte Carlo and Quasi-Monte Carlo Methods 2006 577–589. Springer, Berlin.
    Zentralblatt MATH: 1141.65324
  • [35] Milstein, G. N. (1995). Numerical Integration of Stochastic Differential Equations. Mathematics and Its Applications 313. Kluwer, Dordrecht.
  • [36] Ostermann, A., Thalhammer, M. and Wright, W. M. (2006). A class of explicit exponential general linear methods. BIT 46 409–431.
    Mathematical Reviews (MathSciNet): MR2238680
    Zentralblatt MATH: 1103.65061
    Digital Object Identifier: doi:10.1007/s10543-006-0054-3
  • [37] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Math. 1905. Springer, Berlin.
  • [38] Rößler, A. (2009). Runge–Kutta Methods for the Numerical Solution of Stochastic Differential Equations. Shaker, Aachen.
  • [39] Rößler, A. (2004). Stochastic Taylor expansions for the expectation of functionals of diffusion processes. Stoch. Anal. Appl. 22 1553–1576.
    Zentralblatt MATH: 1065.60068
    Digital Object Identifier: doi:10.1081/SAP-200029495
  • [40] Rößler, A. (2006). Rooted tree analysis for order conditions of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations. Stoch. Anal. Appl. 24 97–134.
  • [41] Sell, G. R. and You, Y. (2002). Dynamics of Evolutionary Equations. Applied Mathematical Sciences 143. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1873467
    Zentralblatt MATH: 1254.37002
  • [42] Shardlow, T. (1999). Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim. 20 121–145.
    Zentralblatt MATH: 0919.65100
    Digital Object Identifier: doi:10.1080/01630569908816884

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