Exponential moments for numerical approximations of stochastic partial differential equations

Article

Abstract

Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it is an important research problem to construct and analyze discrete numerical approximation schemes which converge with strong convergence rates to the solutions of such SPDEs. In the case of finite dimensional stochastic ordinary differential equations (SODEs) with non-globally monotone nonlinearities it has recently been revealed that exponential integrability properties of the discrete numerical approximation scheme are a key instrument to establish strong convergence rates for the considered approximation scheme. Exponential integrability properties for appropriate approximation schemes have been established in the literature in the case of a large class of finite dimensional SODEs. To the best of our knowledge, there exists no result in the scientific literature which proves exponential integrability properties for a time discrete approximation scheme in the case of a SPDE. In particular, to the best of our knowledge, there exists no result in the scientific literature which establishes strong convergence rates for a time discrete approximation scheme in the case of a SPDE with a non-globally monotone nonlinearity. In this paper we propose a new class of tamed space-time-noise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of SPDEs. More specifically, the main result of this article proves that these approximation schemes enjoy exponential integrability properties for a large class of semilinear SPDEs with possibly non-globally monotone nonlinearities. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations, stochastic Kuramoto–Sivashinsky equations, and two-dimensional stochastic Navier–Stokes equations.

Keywords

Stochastic partial differential equation Numerical analysis Lyapunov function Exponential moments Approximation scheme 

Notes

Acknowledgements

We gratefully acknowledge Zdzisław Brzeźniak for several useful comments that helped to improve the presentation of the results. This project has been supported through the SNSF-Research Project \( 200021\_156603 \) “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”.

References

  1. 1.
    Andersson, A., Jentzen, A., Kurniawan, R.: Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. arXiv:1512.06899 (2016)
  2. 2.
    Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework. Texts in Applied Mathematics, vol. 39, 3rd edn. Springer, Dordrecht (2009)MATHGoogle Scholar
  3. 3.
    Becker, S., Jentzen, A.: Strong, convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations. arXiv:1601.05756 (2016)
  4. 4.
    Bessaih, H., Brzeźniak, Z., Millet, A.: Splitting up method for the 2D stochastic Navier–Stokes equations. Stoch. Partial Differ. Equ. Anal. Comput. 2(4), 433–470 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Birnir, B.: The Kolmogorov–Obukhov statistical theory of turbulence. J. Nonlinear Sci. 23(4), 657–688 (2013)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Birnir, B.: The Kolmogorov–Obukhov Theory of Turbulence: A Mathematical Theory of Turbulence. SpringerBriefs in Mathematics. Springer, New York (2013)MATHCrossRefGoogle Scholar
  7. 7.
    Blömker, D., Romito, M.: Stochastic PDEs and lack of regularity: a surface growth equation with noise: existence, uniqueness, and blow-up. Jahresber. Dtsch. Math. Ver. 117(4), 233–286 (2015)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bogachev, V.I.: Measure Theory, vol. I, II. Springer, Berlin (2007)MATHCrossRefGoogle Scholar
  9. 9.
    Bou-Rabee, N., Hairer, M.: Nonasymptotic mixing of the MALA algorithm. IMA J. Numer. Anal. 33(1), 80–110 (2013)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Brzeźniak, Z., Carelli, E., Prohl, A.: Finite-element-based discretizations of the incompressible Navier-Stokes equations with multiplicative random forcing. IMA J. Numer. Anal. 33(3), 771–824 (2013).  https://doi.org/10.1093/imanum/drs032 MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Carelli, E., Hausenblas, E., Prohl, A.: Time-splitting methods to solve the stochastic incompressible Stokes equation. SIAM J. Numer. Anal. 50(6), 2917–2939 (2012)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Carelli, E., Prohl, A.: Rates of convergence for discretizations of the stochastic incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 50(5), 2467–2496 (2012)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Cox, S.G., Hutzenthaler, M., Jentzen, A.: Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations. arXiv:1309.5595v2 (2014)
  14. 14.
    Cozma, A., Reisinger, C.: Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process. Discrete Contin. Dyn. Syst. Ser. B 21(10), 3359–3377 (2016)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Da Prato, G., Debussche, A.: Two-dimensional Navier–Stokes equations driven by a space–time white noise. J. Funct. Anal. 196(1), 180–210 (2002)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers’ equation. NoDEA Nonlinear Differ. Equ. Appl. 1(4), 389–402 (1994)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Da Prato, G., Jentzen, A., Röckner, M.A.: Mild Ito formula for SPDEs. arXiv:1009.3526 (2012)
  18. 18.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1992)MATHCrossRefGoogle Scholar
  19. 19.
    Dörsek, P.: Semigroup splitting and cubature approximations for the stochastic Navier–Stokes equations. SIAM J. Numer. Anal. 50(2), 729–746 (2012)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Duan, J., Ervin, V.J.: On the stochastic Kuramoto–Sivashinsky equation. Nonlinear Anal. 44(2), 205–216 (2001)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Filipović, D., Tappe, S., Teichmann, J.: Term structure models driven by Wiener processes and Poisson measures: existence and positivity. SIAM J. Financ. Math. 1(1), 523–554 (2010)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Gyöngy, I., Krylov, N.: Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Relat. Fields 105(2), 143–158 (1996)MATHCrossRefGoogle Scholar
  23. 23.
    Gyöngy, I., Sabanis, S., Šiška, D.: Convergence of tamed Euler schemes for a class of stochastic evolution equations. Stoch. Partial Differ. Equ. Anal. Comput. 4(2), 225–245 (2016)MathSciNetMATHGoogle Scholar
  24. 24.
    Hairer, M.: Solving the KPZ equation. Ann. Math. 178, 559–664 (2013)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Hairer, M., Voss, J.: Approximations to the stochastic Burgers equation. J. Nonlinear Sci. 21(6), 897–920 (2011)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Harms, P., Stefanovits, D., Teichmann, J., Wüthrich, M.V.: Consistent recalibration of yield curve models. Math. Finance. https://onlinelibrary.wiley.com/doi/abs/10.1111/mafi.12159 (2017)
  27. 27.
    Hutzenthaler, M., Jentzen, A.: On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients. arXiv:1401.0295 (2014)
  28. 28.
    Hutzenthaler, M., Jentzen, A.: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Am. Math. Soc. 236, 1112 (2015)MathSciNetMATHGoogle Scholar
  29. 29.
    Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 22(4), 1611–1641 (2012)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Hutzenthaler, M., Jentzen, A., Salimova, D.: Strong, convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto–Sivashinsky equations. arXiv:1604.02053 (2016)
  31. 31.
    Hutzenthaler, M., Jentzen, A., Wang, X.: Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Math. Comput. 87(311), 1353–1413 (2018)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Jentzen, A., Pušnik, P.: Strong, convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. arXiv:1504.03523 (2015)
  33. 33.
    Jentzen, A., Pušnik, P.: Exponential moments for numerical approximations of stochastic partial differential equations. arXiv:1609.07031 (2016)
  34. 34.
    Kallianpur, G., Xiong, J.: Stochastic models of environmental pollution. Adv. Appl. Probab. 26(2), 377–403 (1994)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Kouritzin, M.A., Long, H.: Convergence of Markov chain approximations to stochastic reaction–diffusion equations. Ann. Appl. Probab. 12(3), 1039–1070 (2002)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Mourrat, J.-C., Weber, H.: Convergence of the two-dimensional dynamic Ising–Kac model to \(\Phi ^4_2\). Commun. Pure Appl. Math. 70(4), 717–812 (2017)MATHGoogle Scholar
  37. 37.
    Parthasarathy, K.R.: Probability Measures on Metric Spaces. Probability and Mathematical Statistics, vol. 3. Academic Press Inc, New York (1967)MATHGoogle Scholar
  38. 38.
    Sabanis, S.: A note on tamed Euler approximations. Electron. Commun. Probab. 18, 1–10 (2013)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Sabanis, S.: Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients. Ann. Appl. Probab. 26(4), 2083–2105 (2016)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Volume of 143 Applied Mathematical Sciences. Springer, New York (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSeminar for Applied MathematicsZurichSwitzerland

Personalised recommendations