Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations

Hutzenthaler, Martin; Jentzen, Arnulf; Wang, Xiaojie

doi:10.1090/mcom/3146

Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations
HTML articles powered by AMS MathViewer

by Martin Hutzenthaler, Arnulf Jentzen and Xiaojie Wang PDF

Math. Comp. 87 (2018), 1353-1413 Request permission

Abstract:

Exponential integrability properties of numerical approximations are a key tool for establishing positive rates of strong and numerically weak convergence for a large class of nonlinear stochastic differential equations. It turns out that well-known numerical approximation processes such as Euler-Maruyama approximations, linear-implicit Euler approximations, and some tamed Euler approximations from the literature rarely preserve exponential integrability properties of the exact solution. The main contribution of this article is to identify a class of stopped increment-tamed Euler approximations which preserve exponential integrability properties of the exact solution under minor additional assumptions on the involved functions.

References

N. Bou-Rabee and M. Hairer, Nonasymptotic mixing of the MALA algorithm, IMA J. Numer. Anal. 33 (2013), no. 1, 80–110. MR 3020951, DOI 10.1093/imanum/drs003
Zdzislaw Brzeźniak, Erich Carelli, and Andreas Prohl, Finite-element-based discretizations of the incompressible Navier-Stokes equations with multiplicative random forcing, IMA J. Numer. Anal. 33 (2013), no. 3, 771–824. MR 3081484, DOI 10.1093/imanum/drs032
S. G. Cox, M. Hutzenthaler, and A. Jentzen, Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations, arXiv:1309.5595 (2014), 1–84.
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
Steffen Dereich, Andreas Neuenkirch, and Lukasz Szpruch, 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 (2012), no. 2140, 1105–1115. MR 2898556, DOI 10.1098/rspa.2011.0505
Philipp Dörsek, Semigroup splitting and cubature approximations for the stochastic Navier-Stokes equations, SIAM J. Numer. Anal. 50 (2012), no. 2, 729–746. MR 2914284, DOI 10.1137/110833841
Abdelhadi Es-Sarhir and Wilhelm Stannat, Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications, J. Funct. Anal. 259 (2010), no. 5, 1248–1272. MR 2652188, DOI 10.1016/j.jfa.2010.02.017
Shizan Fang, Peter Imkeller, and Tusheng Zhang, Global flows for stochastic differential equations without global Lipschitz conditions, Ann. Probab. 35 (2007), no. 1, 180–205. MR 2303947, DOI 10.1214/009117906000000412
István Gyöngy and Annie Millet, On discretization schemes for stochastic evolution equations, Potential Anal. 23 (2005), no. 2, 99–134. MR 2139212, DOI 10.1007/s11118-004-5393-6
Martin Hairer and Jonathan C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math. (2) 164 (2006), no. 3, 993–1032. MR 2259251, DOI 10.4007/annals.2006.164.993
Nikolaos Halidias, A novel approach to construct numerical methods for stochastic differential equations, Numer. Algorithms 66 (2014), no. 1, 79–87. MR 3197358, DOI 10.1007/s11075-013-9724-9
Matthias Hieber and Wilhelm Stannat, Stochastic stability of the Ekman spiral, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 1, 189–208. MR 3088441
Desmond J. Higham, Xuerong Mao, and Andrew M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), no. 3, 1041–1063. MR 1949404, DOI 10.1137/S0036142901389530
Desmond J. Higham, Xuerong Mao, and Lukasz Szpruch, Convergence, non-negativity and stability of a new Milstein scheme with applications to finance, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), no. 8, 2083–2100. MR 3082312, DOI 10.3934/dcdsb.2013.18.2083
Yaozhong Hu, Semi-implicit Euler-Maruyama scheme for stiff stochastic equations, Stochastic analysis and related topics, V (Silivri, 1994) Progr. Probab., vol. 38, Birkhäuser Boston, Boston, MA, 1996, pp. 183–202. MR 1396331
Martin Hutzenthaler and Arnulf Jentzen, Convergence of the stochastic Euler scheme for locally Lipschitz coefficients, Found. Comput. Math. 11 (2011), no. 6, 657–706. MR 2859952, DOI 10.1007/s10208-011-9101-9
M. Hutzenthaler and A. Jentzen, 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), 1–41.
Martin Hutzenthaler and Arnulf Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc. 236 (2015), no. 1112, v+99. MR 3364862, DOI 10.1090/memo/1112
Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden, 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 (2011), no. 2130, 1563–1576. MR 2795791, DOI 10.1098/rspa.2010.0348
Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab. 22 (2012), no. 4, 1611–1641. MR 2985171, DOI 10.1214/11-AAP803
Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden, Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations, Ann. Appl. Probab. 23 (2013), no. 5, 1913–1966. MR 3134726, DOI 10.1214/12-AAP890
Achim Klenke, Probability theory, Universitext, Springer-Verlag London, Ltd., London, 2008. A comprehensive course; Translated from the 2006 German original. MR 2372119, DOI 10.1007/978-1-84800-048-3
Xue-Mei Li, Strong $p$-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds, Probab. Theory Related Fields 100 (1994), no. 4, 485–511. MR 1305784, DOI 10.1007/BF01268991
Wei Liu and Xuerong Mao, Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, Appl. Math. Comput. 223 (2013), 389–400. MR 3116272, DOI 10.1016/j.amc.2013.08.023
Xuerong Mao and Lukasz Szpruch, Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math. 238 (2013), 14–28. MR 2972586, DOI 10.1016/j.cam.2012.08.015
G. N. Milstein, E. Platen, and H. Schurz, Balanced implicit methods for stiff stochastic systems, SIAM J. Numer. Anal. 35 (1998), no. 3, 1010–1019. MR 1619926, DOI 10.1137/S0036142994273525
G. N. Milstein and M. V. Tretyakov, Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients, SIAM J. Numer. Anal. 43 (2005), no. 3, 1139–1154. MR 2177799, DOI 10.1137/040612026
Andreas Neuenkirch and Lukasz Szpruch, First order strong approximations of scalar SDEs defined in a domain, Numer. Math. 128 (2014), no. 1, 103–136. MR 3248050, DOI 10.1007/s00211-014-0606-4
Claudia Prévôt and Michael Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007. MR 2329435
Sotirios Sabanis, A note on tamed Euler approximations, Electron. Commun. Probab. 18 (2013), no. 47, 10. MR 3070913, DOI 10.1214/ECP.v18-2824
Sotirios Sabanis, Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients, Ann. Appl. Probab. 26 (2016), no. 4, 2083–2105. MR 3543890, DOI 10.1214/15-AAP1140
Henri Schurz, General theorems for numerical approximation of stochastic processes on the Hilbert space $H_2([0,T],\mu ,\Bbb R^d)$, Electron. Trans. Numer. Anal. 16 (2003), 50–69. MR 1988720
Henri Schurz, Convergence and stability of balanced implicit methods for systems of SDEs, Int. J. Numer. Anal. Model. 2 (2005), no. 2, 197–220. MR 2111748
Lukasz Szpruch, Xuerong Mao, Desmond J. Higham, and Jiazhu Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT 51 (2011), no. 2, 405–425. MR 2806537, DOI 10.1007/s10543-010-0288-y
M. V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal. 51 (2013), no. 6, 3135–3162. MR 3129758, DOI 10.1137/120902318
Xiaojie Wang and Siqing Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Difference Equ. Appl. 19 (2013), no. 3, 466–490. MR 3037286, DOI 10.1080/10236198.2012.656617
Xicheng Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stochastic Process. Appl. 120 (2010), no. 10, 1929–1949. MR 2673982, DOI 10.1016/j.spa.2010.05.015