• Previous Article
    Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function
  • DCDS-B Home
  • This Issue
  • Next Article
    Convergence analysis of a symplectic semi-discretization for stochastic nls equation with quadratic potential
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  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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar
[32]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar
[32]

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

[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. Google Scholar

[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. Google Scholar

[1]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027

[2]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[3]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic & Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53

[4]

Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097

[5]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[6]

Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic & Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034

[7]

Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282

[8]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481

[9]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019061

[10]

Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071

[11]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503

[12]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[13]

Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001

[14]

Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057

[15]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279

[16]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[17]

Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47

[18]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

[19]

Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331

[20]

Linlin Tian, Xiaoyi Zhang, Yizhou Bai. Optimal dividend of compound poisson process under a stochastic interest rate. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019047

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (32)
  • HTML views (198)
  • Cited by (0)

Other articles
by authors

[Back to Top]