Annals of Probability

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

Martin Hutzenthaler and Arnulf Jentzen

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Abstract

We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $L^{p}$-distance between the solution process of an SDE and an arbitrary Itô process, which we view as a perturbation of the solution process of the SDE, by the $L^{q}$-distances of the differences of the local characteristics for suitable $p,q>0$. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

Article information

Source
Ann. Probab., Volume 48, Number 1 (2020), 53-93.

Dates
Received: September 2018
First available in Project Euclid: 25 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aop/1585123323

Digital Object Identifier
doi:10.1214/19-AOP1345

Mathematical Reviews number (MathSciNet)
MR4079431

Zentralblatt MATH identifier
07206753

Subjects
Primary: 65C30: Stochastic differential and integral equations

Keywords
Perturbation stochastic differential equation convergence rate nonglobally monotone small-noise analysis Cahn–Hilliard–Cook equation stochastic Burgers equation

Citation

Hutzenthaler, Martin; Jentzen, Arnulf. On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients. Ann. Probab. 48 (2020), no. 1, 53--93. doi:10.1214/19-AOP1345. https://projecteuclid.org/euclid.aop/1585123323


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References

  • [1] Alabert, A. and Gyöngy, I. (2006). On numerical approximation of stochastic Burgers’ equation. In From Stochastic Calculus to Mathematical Finance 1–15. Springer, Berlin.
  • [2] Albeverio, S. and Röckner, M. (1991). Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probab. Theory Related Fields 89 347–386.
  • [3] Alfonsi, A. (2013). Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process. Statist. Probab. Lett. 83 602–607.
  • [4] Blömker, D., Kamrani, M. and Hosseini, S. M. (2013). Full discretization of the stochastic Burgers equation with correlated noise. IMA J. Numer. Anal. 33 825–848.
  • [5] Bou-Rabee, N. and Hairer, M. (2013). Nonasymptotic mixing of the MALA algorithm. IMA J. Numer. Anal. 33 80–110.
  • [6] Brzeźniak, Z., Carelli, E. and Prohl, A. (2013). Finite-element-based discretizations of the incompressible Navier–Stokes equations with multiplicative random forcing. IMA J. Numer. Anal. 33 771–824.
  • [7] Carelli, E. and Prohl, A. (2012). Rates of convergence for discretizations of the stochastic incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 50 2467–2496.
  • [8] Cerrai, S. (1998). Differentiability with respect to initial datum for solutions of SPDE’s with no Fréchet differentiable drift term. Commun. Appl. Anal. 2 249–270.
  • [9] Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach. Lecture Notes in Math. 1762. Springer, Berlin.
  • [10] Cerrai, S. (2003). Stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 271–304.
  • [11] Conus, D., Jentzen, A. and Kurniawan, R. (2019). Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. Ann. Appl. Probab. 29 653–716.
  • [12] Cox, S., Hutzenthaler, M., Jentzen, A., van Neerven, J. and Welti, T. (2016). Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions. IMA J. Numer. Anal.. To appear. Available at arXiv:1605.00856.
  • [13] Cox, S. G., Hutzenthaler, M. and Jentzen, A. (2014). Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations. Revision requested from Mem. Amer. Math. Soc.. Available at arXiv:1309.5595v2.
  • [14] Da Prato, G. and Debussche, A. (1996). Stochastic Cahn–Hilliard equation. Nonlinear Anal. 26 241–263.
  • [15] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [16] Datta, S. and Bhattacharjee, J. K. (2001). Effect of stochastic forcing on the Duffing oscillator. Phys. Lett. A 283 323–326.
  • [17] 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.
  • [18] Dereich, S., Neuenkirch, A. and Szpruch, L. (2012). An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 1105–1115.
  • [19] Dörsek, P. (2012). Semigroup splitting and cubature approximations for the stochastic Navier–Stokes equations. SIAM J. Numer. Anal. 50 729–746.
  • [20] Es-Sarhir, A. and Stannat, W. (2010). Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications. J. Funct. Anal. 259 1248–1272.
  • [21] Fang, S., Imkeller, P. and Zhang, T. (2007). Global flows for stochastic differential equations without global Lipschitz conditions. Ann. Probab. 35 180–205.
  • [22] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, Heidelberg.
  • [23] Giles, M. (2008). Improved multilevel Monte Carlo convergence using the Milstein scheme. In Monte Carlo and Quasi-Monte Carlo Methods 2006 343–358. Springer, Berlin.
  • [24] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
  • [25] Gyöngy, I. and Millet, A. (2005). On discretization schemes for stochastic evolution equations. Potential Anal. 23 99–134.
  • [26] Gyöngy, I. and Rásonyi, M. (2011). A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stochastic Process. Appl. 121 2189–2200.
  • [27] Hairer, M., Hutzenthaler, M. and Jentzen, A. (2015). Loss of regularity for Kolmogorov equations. Ann. Probab. 43 468–527.
  • [28] Hairer, M. and Mattingly, J. C. (2006). Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 993–1032.
  • [29] Heinrich, S. (1998). Monte Carlo complexity of global solution of integral equations. J. Complexity 14 151–175.
  • [30] Heinrich, S. (2001). Multilevel Monte Carlo methods. In Large-Scale Scientific Computing. Lecture Notes Comput. Sci. 2179 58–67. Springer, Berlin.
  • [31] Hieber, M. and Stannat, W. (2013). Stochastic stability of the Ekman spiral. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 189–208.
  • [32] Higham, D. J., Mao, X. and Stuart, A. M. (2002). Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40 1041–1063.
  • [33] Hu, Y. (1996). Semi-implicit Euler–Maruyama scheme for stiff stochastic equations. In Stochastic Analysis and Related Topics, V (Silivri, 1994). Progress in Probability 38 183–202. Birkhäuser, Boston, MA.
  • [34] Hutzenthaler, M. and Jentzen, A. (2015). Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Amer. Math. Soc. 236 v $+$ 99.
  • [35] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2011). Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 1563–1576.
  • [36] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2012). Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 1611–1641.
  • [37] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2013). Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Ann. Appl. Probab. 23 1913–1966.
  • [38] Hutzenthaler, M., Jentzen, A. and Wang, X. (2018). Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Math. Comp. 87 1353–1413.
  • [39] Ichikawa, A. (1984). Semilinear stochastic evolution equations: Boundedness, stability and invariant measures. Stochastics 12 1–39.
  • [40] Jacobe de Naurois, L., Jentzen, A. and Welti, T. (2018). Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations. In Stochastic Partial Differential Equations and Related Fields. Springer Proc. Math. Stat. 229 237–248. Springer, Cham.
  • [41] Jentzen, A. and Pušnik, P. (2015). Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. IMA J. Numer. Anal. To appear. Available at arXiv:1504.03523.
  • [42] Kamrani, M. and Blömker, D. (2017). Pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local Lipschitz condition. J. Comput. Appl. Math. 323 123–135.
  • [43] Kebaier, A. (2005). Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15 2681–2705.
  • [44] Kloeden, P. and Neuenkirch, A. (2013). Convergence of numerical methods for stochastic differential equations in mathematical finance. In Recent Developments in Computational Finance. Interdiscip. Math. Sci. 14 49–80. World Scientific, Hackensack, NJ.
  • [45] Kovács, M., Larsson, S. and Lindgren, F. (2015). On the backward Euler approximation of the stochastic Allen–Cahn equation. J. Appl. Probab. 52 323–338.
  • [46] Kovács, M., Larsson, S. and Mesforush, A. (2011). Finite element approximation of the Cahn–Hilliard–Cook equation. SIAM J. Numer. Anal. 49 2407–2429.
  • [47] Kühn, C. (2004). Stochastische Analysis mit Finanzmathematik.
  • [48] Leha, G. and Ritter, G. (1994). Lyapunov-type conditions for stationary distributions of diffusion processes on Hilbert spaces. Stoch. Stoch. Rep. 48 195–225.
  • [49] Leha, G. and Ritter, G. (2003). Lyapunov functions and stationary distributions of stochastic evolution equations. Stoch. Anal. Appl. 21 763–799.
  • [50] Li, X.-M. (1994). Strong $p$-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds. Probab. Theory Related Fields 100 485–511.
  • [51] Liu, D. (2003). Convergence of the spectral method for stochastic Ginzburg–Landau equation driven by space–time white noise. Commun. Math. Sci. 1 361–375.
  • [52] Mao, X. and Szpruch, L. (2013). Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients. Stochastics 85 144–171.
  • [53] Maslowski, B. (1986). On some stability properties of stochastic differential equations of Itô’s type. Čas. Pěst. Mat. 111 404–423, 435.
  • [54] Minty, G. J. (1962). Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29 341–346.
  • [55] Minty, G. J. (1963). On a “monotonicity” method for the solution of non-linear equations in Banach spaces. Proc. Natl. Acad. Sci. USA 50 1038–1041.
  • [56] 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.
  • [57] 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.
  • [58] Müller-Gronbach, T., Ritter, K. and Wagner, T. (2008). Optimal pointwise approximation of infinite-dimensional Ornstein–Uhlenbeck processes. Stoch. Dyn. 8 519–541.
  • [59] Neuenkirch, A. and Szpruch, L. (2014). First order strong approximations of scalar SDEs defined in a domain. Numer. Math. 128 103–136.
  • [60] Pardoux, E. (1975). Équations aux dérivées partielles stochastiques de type monotone. In Séminaire sur les Équations aux Dérivées Partielles (19741975), III, Exp. No. 2 1–10.
  • [61] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Math. 1905. Springer, Berlin.
  • [62] Printems, J. (2001). On the discretization in time of parabolic stochastic partial differential equations. ESAIM Math. Model. Numer. Anal. 35 1055–1078.
  • [63] Sabanis, S. (2013). A note on tamed Euler approximations. Electron. Commun. Probab. 18 Art. ID 47.
  • [64] Sabanis, S. (2016). Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients. Ann. Appl. Probab. 26 2083–2105.
  • [65] Sauer, M. and Stannat, W. (2015). Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition. Math. Comp. 84 743–766.
  • [66] Schenk-Hoppé, K. R. (1996). Deterministic and stochastic Duffing–van der Pol oscillators are non-explosive. Z. Angew. Math. Phys. 47 740–759.
  • [67] Sell, G. R. and You, Y. (2002). Dynamics of Evolutionary Equations. Applied Mathematical Sciences 143. Springer, New York.
  • [68] Szpruch, L. (2013). V-stable tamed Euler schemes. Available at arXiv:1310.0785.
  • [69] Tretyakov, M. V. and Zhang, Z. (2013). A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications. SIAM J. Numer. Anal. 51 3135–3162.
  • [70] Zhang, X. (2010). Stochastic flows and Bismut formulas for stochastic Hamiltonian systems. Stochastic Process. Appl. 120 1929–1949.
  • [71] Zhou, X. and E, W. (2010). Study of noise-induced transitions in the Lorenz system using the minimum action method. Commun. Math. Sci. 8 341–355.