## On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients.(English)Zbl 07206753

Summary: 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.

### MSC:

 65C30 Numerical solutions to stochastic differential and integral equations
Full Text:

### 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. · Zbl 1116.60027 [2] Albeverio, S. and Röckner, M. (1991). Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probab. Theory Related Fields 89 347-386. · Zbl 0725.60055 [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. · Zbl 1273.65007 [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. · Zbl 1280.65008 [5] Bou-Rabee, N. and Hairer, M. (2013). Nonasymptotic mixing of the MALA algorithm. IMA J. Numer. Anal. 33 80-110. · Zbl 1305.65012 [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. · Zbl 1426.76227 [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. · Zbl 1426.76231 [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. · Zbl 0897.60068 [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. · Zbl 1027.60064 [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. · Zbl 1477.65020 [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. · Zbl 1460.65054 [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. · Zbl 0838.60056 [15] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge. · Zbl 0761.60052 [16] Datta, S. and Bhattacharjee, J. K. (2001). Effect of stochastic forcing on the Duffing oscillator. Phys. Lett. A 283 323-326. · Zbl 1008.70018 [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. · Zbl 0956.60064 [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. · Zbl 1364.65013 [19] Dörsek, P. (2012). Semigroup splitting and cubature approximations for the stochastic Navier-Stokes equations. SIAM J. Numer. Anal. 50 729-746. · Zbl 1247.60090 [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. · Zbl 1202.60097 [21] Fang, S., Imkeller, P. and Zhang, T. (2007). Global flows for stochastic differential equations without global Lipschitz conditions. Ann. Probab. 35 180-205. · Zbl 1128.60046 [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. · Zbl 1267.60004 [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. · Zbl 1141.65321 [24] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607-617. · Zbl 1167.65316 [25] Gyöngy, I. and Millet, A. (2005). On discretization schemes for stochastic evolution equations. Potential Anal. 23 99-134. · Zbl 1067.60049 [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. · Zbl 1226.60095 [27] Hairer, M., Hutzenthaler, M. and Jentzen, A. (2015). Loss of regularity for Kolmogorov equations. Ann. Probab. 43 468-527. · Zbl 1322.35083 [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. · Zbl 1130.37038 [29] Heinrich, S. (1998). Monte Carlo complexity of global solution of integral equations. J. Complexity 14 151-175. · Zbl 0920.65090 [30] Heinrich, S. (2001). Multilevel Monte Carlo methods. In Large-Scale Scientific Computing. Lecture Notes Comput. Sci. 2179 58-67. Springer, Berlin. · Zbl 1031.65005 [31] Hieber, M. and Stannat, W. (2013). Stochastic stability of the Ekman spiral. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 189-208. · Zbl 1264.35293 [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. · Zbl 1026.65003 [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. · Zbl 0848.60057 [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 $$+ 99$$. · Zbl 1330.60084 [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. · Zbl 1228.65014 [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. · Zbl 1256.65003 [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. · Zbl 1283.60098 [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. · Zbl 1432.65011 [39] Ichikawa, A. (1984). Semilinear stochastic evolution equations: Boundedness, stability and invariant measures. Stochastics 12 1-39. · Zbl 0538.60068 [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. · Zbl 1405.60094 [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. · Zbl 1466.65161 [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. · Zbl 1364.60088 [43] Kebaier, A. (2005). Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15 2681-2705. · Zbl 1099.65011 [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. · Zbl 1277.91194 [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. · Zbl 1323.60089 [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. · Zbl 1248.65012 [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. · Zbl 0828.60063 [49] Leha, G. and Ritter, G. (2003). Lyapunov functions and stationary distributions of stochastic evolution equations. Stoch. Anal. Appl. 21 763-799. · Zbl 1029.60049 [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. · Zbl 0815.60050 [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. · Zbl 1086.60037 [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. · Zbl 1304.65009 [53] Maslowski, B. (1986). On some stability properties of stochastic differential equations of Itô’s type. Čas. Pěst. Mat. 111 404-423, 435. · Zbl 0625.60066 [54] Minty, G. J. (1962). Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29 341-346. · Zbl 0111.31202 [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. · Zbl 0124.07303 [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. · Zbl 1136.60044 [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. · Zbl 1141.65324 [58] Müller-Gronbach, T., Ritter, K. and Wagner, T. (2008). Optimal pointwise approximation of infinite-dimensional Ornstein-Uhlenbeck processes. Stoch. Dyn. 8 519-541. · Zbl 1153.60371 [59] Neuenkirch, A. and Szpruch, L. (2014). First order strong approximations of scalar SDEs defined in a domain. Numer. Math. 128 103-136. · Zbl 1306.60075 [60] Pardoux, E. (1975). Équations aux dérivées partielles stochastiques de type monotone. In Séminaire sur les Équations aux Dérivées Partielles (1974-1975), 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. · Zbl 1123.60001 [62] Printems, J. (2001). On the discretization in time of parabolic stochastic partial differential equations. ESAIM Math. Model. Numer. Anal. 35 1055-1078. · Zbl 0991.60051 [63] Sabanis, S. (2013). A note on tamed Euler approximations. Electron. Commun. Probab. 18 Art. ID 47. · Zbl 1329.60237 [64] Sabanis, S. (2016). Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients. Ann. Appl. Probab. 26 2083-2105. · Zbl 1352.60101 [65] Sauer, M. and Stannat, W. (2015). Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition. Math. Comp. 84 743-766. · Zbl 1337.60162 [66] Schenk-Hoppé, K. R. (1996). Deterministic and stochastic Duffing-van der Pol oscillators are non-explosive. Z. Angew. Math. Phys. 47 740-759. · Zbl 0876.70018 [67] Sell, G. R. and You, Y. (2002). Dynamics of Evolutionary Equations. Applied Mathematical Sciences 143. Springer, New York. · Zbl 1254.37002 [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. · Zbl 1293.60069 [70] Zhang, X. (2010). Stochastic flows and Bismut formulas for stochastic Hamiltonian systems. Stochastic Process. Appl. 120 1929-1949. · Zbl 1200.60049 [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. · Zbl 1202.34104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.