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August  2019, 24(8): 4271-4294. doi: 10.3934/dcdsb.2019081

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

1. 

Smarter AI Sweden, Vallgatan 3, SE-411-16 Gothenburg, Sweden

2. 

Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany

* Corresponding author: Felix Lindner

Received  June 2018 Revised  November 2018 Published  August 2019 Early access  April 2019

We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable Lévy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.

Citation: Adam Andersson, Felix Lindner. Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4271-4294. doi: 10.3934/dcdsb.2019081
References:
[1]

A. AnderssonM. Kovács and S. Larsson, Weak and strong error analysis for semilinear stochastic Volterra equations, J. Math. Anal. Appl., 437 (2016), 1283-1304.  doi: 10.1016/j.jmaa.2015.09.016.

[2]

A. AnderssonR. Kruse and S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximation of SPDE, Stoch. PDE: Anal. Comp., 4 (2016), 113-149.  doi: 10.1007/s40072-015-0065-7.

[3]

A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation, Math. Comp., 85 (2016), 1335-1358.  doi: 10.1090/mcom/3016.

[4]

A. Andersson and F. Lindner, Poisson Malliavin calculus in Hilbert space with an application to SPDEs, preprint, arXiv: 1703.07259.

[5]

A. Barth and A. Lang, Simulation of stochastic partial differential equations using finite element methods, Stochastics, 84 (2012), 217-231.  doi: 10.1080/17442508.2010.523466.

[6]

A. Barth and A. Stein, Approximation and simulation of infinite-dimensional Lévy processes, Stoch. PDE: Anal. Comp., 6 (2018), 286-334.  doi: 10.1007/s40072-017-0109-2.

[7]

A. Barth and T. Stüwe, Weak convergence of Galerkin approximations of stochastic partial differential equations driven by additive Lévy noise, Math. Comput. Simulation, 143 (2018), 215-225.  doi: 10.1016/j.matcom.2017.03.007.

[8]

B. Birnir, The Kolmogorov-Obukhov Statistical Theory of Turbulence, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-6262-0.

[9]

C.-E. Bréhier, Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs, preprint, arXiv: 1709.09370.

[10]

C.-E. Bréhier and A. Debussche, Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient, J. Math. Pures Appl., 119 (2018), 193-254.  doi: 10.1016/j.matpur.2018.08.010.

[11]

C.-E. Bréhier and L. Goudenège, Weak convergence rates of splitting schemes for the stochastic Allen-Cahn equation, preprint, arXiv: 1804.04061.

[12]

C.-E. BréhierM. Hairer and A. M. Stuart, Weak error estimates for trajectories of SPDEs under spectral Galerkin discretization, J. Comput. Math., 36 (2018), 159-182.  doi: 10.4208/jcm.1607-m2016-0539.

[13]

B. Chen, C. Chong and C. Klüppelberg, Simulation of stochastic Volterra equations driven by space-time Lévy noise, The Fascination of Probability, Statistics and their Applications, (2015), 209–229. doi: 10.1007/978-3-319-25826-3_10.

[14]

D. Conus, A. Jentzen and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, Ann. Appl. Probab., 29 (2019), 653–716, arXiv: 1408.1108v2. doi: 10.1214/17-AAP1352.

[15]

A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp., 80 (2011), 89-117.  doi: 10.1090/S0025-5718-2010-02395-6.

[16]

T. DunstE. Hausenblas and A. Prohl, Approximate Euler method for parabolic stochastic partial differential equations driven by space-time Lévy noise, SIAM J. Numer. Anal., 50 (2012), 2873-2896.  doi: 10.1137/100818297.

[17]

C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp., 58 (1992), 603-630.  doi: 10.1090/S0025-5718-1992-1122067-1.

[18]

E. Hausenblas, Weak approximation of the stochastic wave equation, J. Comput. Appl. Math., 235 (2010), 33-58.  doi: 10.1016/j.cam.2010.03.026.

[19]

M. Hefter, A. Jentzen and R. Kurniawan, Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces, preprint, arXiv: 1612.03209.

[20]

L. Jacobe de Naurois, A. Jentzen and T. Welti, Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations, Stochastic Partial Differential Equations and Related Fields, 237–248, Springer Proc. Math. Stat., 229, Springer, Cham, 2018, arXiv: 1701.04351.

[21]

L. Jacobe de Naurois, A. Jentzen and T. Welti, Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise, preprint, accepted in Appl. Math. Optim., arXiv: 1508.05168.

[22]

A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients, preprint, arXiv: 1501.03539.

[23]

G. Kallianpur and J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, vol. 26 of Institute of Mathematical Statistics, Lecture Notes, SIAM, Philadelphia, 1995.

[24]

K. KirchnerA. Lang and S. Larsson, Covariance structure of parabolic stochastic partial differential equations with multiplicative Lévy noise, J. Differential Equations, 262 (2017), 5896-5927.  doi: 10.1016/j.jde.2017.02.021.

[25]

M. KovácsF. Lindner and R. L. Schilling, Weak convergence of finite element approximations of linear stochastic evolution equations with additive Lévy noise, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 1159-1199.  doi: 10.1137/15M1009792.

[26]

R. Kruse, Strong and Weak Approximation of Stochastic Evolution Equations, vol. 2093 of Lecture Notes in Math., Springer, 2014.

[27]

A. Lang and A. Petersson, Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations, Math. Comput. Simulation, 143 (2018), 99-113.  doi: 10.1016/j.matcom.2017.05.002.

[28]

G. Last, Stochastic analysis for Poisson processes, in Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry (eds. G. Peccati and M. Reitzner), Springer International Publishing, Cham, 7 (2016), 1–36. doi: 10.1007/978-3-319-05233-5_1.

[29]

F. Lindner and R. L. Schilling, Weak order for the discretization of the stochastic heat equation driven by impulsive noise, Potential Anal., 38 (2013), 345-379.  doi: 10.1007/s11118-012-9276-y.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511721373.
[32]

J. Picard, Formules de dualité sur l'espace de Poisson, Ann. Inst. Henri Poincaré Probab. Stat., 32 (1996), 509-548. 

[33]

X. Wang, Weak error estimates of the exponential euler scheme for semi-linear spdes without malliavin calculus, Discrete Contin. Dyn. Syst., 36 (2016), 481-497.  doi: 10.3934/dcds.2016.36.481.

[34]

X. Wang and S. Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise, J. Math. Anal. Appl., 398 (2013), 151-169.  doi: 10.1016/j.jmaa.2012.08.038.

show all references

References:
[1]

A. AnderssonM. Kovács and S. Larsson, Weak and strong error analysis for semilinear stochastic Volterra equations, J. Math. Anal. Appl., 437 (2016), 1283-1304.  doi: 10.1016/j.jmaa.2015.09.016.

[2]

A. AnderssonR. Kruse and S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximation of SPDE, Stoch. PDE: Anal. Comp., 4 (2016), 113-149.  doi: 10.1007/s40072-015-0065-7.

[3]

A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation, Math. Comp., 85 (2016), 1335-1358.  doi: 10.1090/mcom/3016.

[4]

A. Andersson and F. Lindner, Poisson Malliavin calculus in Hilbert space with an application to SPDEs, preprint, arXiv: 1703.07259.

[5]

A. Barth and A. Lang, Simulation of stochastic partial differential equations using finite element methods, Stochastics, 84 (2012), 217-231.  doi: 10.1080/17442508.2010.523466.

[6]

A. Barth and A. Stein, Approximation and simulation of infinite-dimensional Lévy processes, Stoch. PDE: Anal. Comp., 6 (2018), 286-334.  doi: 10.1007/s40072-017-0109-2.

[7]

A. Barth and T. Stüwe, Weak convergence of Galerkin approximations of stochastic partial differential equations driven by additive Lévy noise, Math. Comput. Simulation, 143 (2018), 215-225.  doi: 10.1016/j.matcom.2017.03.007.

[8]

B. Birnir, The Kolmogorov-Obukhov Statistical Theory of Turbulence, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-6262-0.

[9]

C.-E. Bréhier, Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs, preprint, arXiv: 1709.09370.

[10]

C.-E. Bréhier and A. Debussche, Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient, J. Math. Pures Appl., 119 (2018), 193-254.  doi: 10.1016/j.matpur.2018.08.010.

[11]

C.-E. Bréhier and L. Goudenège, Weak convergence rates of splitting schemes for the stochastic Allen-Cahn equation, preprint, arXiv: 1804.04061.

[12]

C.-E. BréhierM. Hairer and A. M. Stuart, Weak error estimates for trajectories of SPDEs under spectral Galerkin discretization, J. Comput. Math., 36 (2018), 159-182.  doi: 10.4208/jcm.1607-m2016-0539.

[13]

B. Chen, C. Chong and C. Klüppelberg, Simulation of stochastic Volterra equations driven by space-time Lévy noise, The Fascination of Probability, Statistics and their Applications, (2015), 209–229. doi: 10.1007/978-3-319-25826-3_10.

[14]

D. Conus, A. Jentzen and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, Ann. Appl. Probab., 29 (2019), 653–716, arXiv: 1408.1108v2. doi: 10.1214/17-AAP1352.

[15]

A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp., 80 (2011), 89-117.  doi: 10.1090/S0025-5718-2010-02395-6.

[16]

T. DunstE. Hausenblas and A. Prohl, Approximate Euler method for parabolic stochastic partial differential equations driven by space-time Lévy noise, SIAM J. Numer. Anal., 50 (2012), 2873-2896.  doi: 10.1137/100818297.

[17]

C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp., 58 (1992), 603-630.  doi: 10.1090/S0025-5718-1992-1122067-1.

[18]

E. Hausenblas, Weak approximation of the stochastic wave equation, J. Comput. Appl. Math., 235 (2010), 33-58.  doi: 10.1016/j.cam.2010.03.026.

[19]

M. Hefter, A. Jentzen and R. Kurniawan, Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces, preprint, arXiv: 1612.03209.

[20]

L. Jacobe de Naurois, A. Jentzen and T. Welti, Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations, Stochastic Partial Differential Equations and Related Fields, 237–248, Springer Proc. Math. Stat., 229, Springer, Cham, 2018, arXiv: 1701.04351.

[21]

L. Jacobe de Naurois, A. Jentzen and T. Welti, Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise, preprint, accepted in Appl. Math. Optim., arXiv: 1508.05168.

[22]

A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients, preprint, arXiv: 1501.03539.

[23]

G. Kallianpur and J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, vol. 26 of Institute of Mathematical Statistics, Lecture Notes, SIAM, Philadelphia, 1995.

[24]

K. KirchnerA. Lang and S. Larsson, Covariance structure of parabolic stochastic partial differential equations with multiplicative Lévy noise, J. Differential Equations, 262 (2017), 5896-5927.  doi: 10.1016/j.jde.2017.02.021.

[25]

M. KovácsF. Lindner and R. L. Schilling, Weak convergence of finite element approximations of linear stochastic evolution equations with additive Lévy noise, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 1159-1199.  doi: 10.1137/15M1009792.

[26]

R. Kruse, Strong and Weak Approximation of Stochastic Evolution Equations, vol. 2093 of Lecture Notes in Math., Springer, 2014.

[27]

A. Lang and A. Petersson, Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations, Math. Comput. Simulation, 143 (2018), 99-113.  doi: 10.1016/j.matcom.2017.05.002.

[28]

G. Last, Stochastic analysis for Poisson processes, in Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry (eds. G. Peccati and M. Reitzner), Springer International Publishing, Cham, 7 (2016), 1–36. doi: 10.1007/978-3-319-05233-5_1.

[29]

F. Lindner and R. L. Schilling, Weak order for the discretization of the stochastic heat equation driven by impulsive noise, Potential Anal., 38 (2013), 345-379.  doi: 10.1007/s11118-012-9276-y.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511721373.
[32]

J. Picard, Formules de dualité sur l'espace de Poisson, Ann. Inst. Henri Poincaré Probab. Stat., 32 (1996), 509-548. 

[33]

X. Wang, Weak error estimates of the exponential euler scheme for semi-linear spdes without malliavin calculus, Discrete Contin. Dyn. Syst., 36 (2016), 481-497.  doi: 10.3934/dcds.2016.36.481.

[34]

X. Wang and S. Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise, J. Math. Anal. Appl., 398 (2013), 151-169.  doi: 10.1016/j.jmaa.2012.08.038.

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