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$ V$-integrability, asymptotic stability and comparison property of explicit numerical schemes for non-linear SDEs


Authors: Łukasz Szpruch and Xīlíng Zhāng
Journal: Math. Comp. 87 (2018), 755-783
MSC (2010): Primary 65C30, 65C05
DOI: https://doi.org/10.1090/mcom/3219
Published electronically: August 3, 2017
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Abstract: Khasminski [Stochastic Stability of Differential Equations, Kluwer Academic Publishers, 1980] showed that the asymptotic stability and the integrability of solutions to stochastic differential equations (SDEs) can be obtained via Lyapunov functions. These properties are, however, not necessarily inherited by standard numerical approximations. In this article we introduce a general class of explicit numerical approximations that are amenable to Khasminski's techniques and are particularly suited for non-globally Lipschitz coefficients. We derive general conditions under which these numerical schemes are bounded in expectation with respect to certain Lyapunov functions, and/or inherit the asymptotic stability of the SDEs. Finally we show that by truncating the noise it is possible to recover the comparison theorem for numerical approximations of non-linear scalar SDEs.


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  • [1] J.-F. Chassagneux, A. Jacquier, and I. Mihaylov, An explicit Euler scheme with strong rate of convergence for non-Lipschitz sdes, arXiv:1405.3561 (2014).
  • [2] 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, https://doi.org/10.3934/dcdsb.2013.18.2083
  • [3] Desmond J. Higham, $ A$-stability and stochastic mean-square stability, BIT 40 (2000), no. 2, 404-409. MR 1765744, https://doi.org/10.1023/A:1022355410570
  • [4] Desmond J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal. 38 (2000), no. 3, 753-769. MR 1781202, https://doi.org/10.1137/S003614299834736X
  • [5] Desmond J. Higham, Xuerong Mao, and Andrew M. Stuart, Exponential mean-square stability of numerical solutions to stochastic differential equations, LMS J. Comput. Math. 6 (2003), 297-313. MR 2051587, https://doi.org/10.1112/S1461157000000462
  • [6] 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, https://doi.org/10.1137/S0036142901389530
  • [7] Desmond J. Higham, Xuerong Mao, and Chenggui Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal. 45 (2007), no. 2, 592-609. MR 2300289, https://doi.org/10.1137/060658138
  • [8] 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, https://doi.org/10.1098/rspa.2010.0348
  • [9] 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, https://doi.org/10.1214/11-AAP803
  • [10] 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).
  • [11] 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, https://doi.org/10.1090/memo/1112
  • [12] M. Hutzenthaler, A. Jentzen, and X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations, arXiv:1309.7657 (2014).
  • [13] Ioannis Karatzas and Steven E. Shreve, Browian Motion and Stochastic Calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940
  • [14] R. Z. Khasminski, Stochastic Stability of Differential Equations, Kluwer Academic Publishers, 1980.
  • [15] Peter E. Kloeden and Eckhard Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374
  • [16] R. Sh. Liptser and A. N. Shiryayev, Theory of Martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1022664
  • [17] X. Mao, Stability of stochastic differential equations with respect to semimartingales, Pitman Research Notes in Mathematics Series, vol. 251, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR 1110584
  • [18] Xuerong Mao, Stochastic versions of the LaSalle theorem, J. Differential Equations 153 (1999), no. 1, 175-195. MR 1682267, https://doi.org/10.1006/jdeq.1998.3552
  • [19] X. Mao, Stochastic Differential Equations and Applications, Horwood Pub Ltd, 2007.
  • [20] Xuerong Mao and Lukasz Szpruch, Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics 85 (2013), no. 1, 144-171. MR 3011916, https://doi.org/10.1080/17442508.2011.651213
  • [21] 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, https://doi.org/10.1016/j.cam.2012.08.015
  • [22] J. C. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101 (2002), no. 2, 185-232. MR 1931266, https://doi.org/10.1016/S0304-4149(02)00150-3
  • [23] G. N. Milstein and M. V. Tretyakov, Stochastic numerics for mathematical physics, Scientific Computation, Springer-Verlag, Berlin, 2004. MR 2069903
  • [24] Sotirios Sabanis, A note on tamed Euler approximations, Electron. Commun. Probab. 18 (2013), no. 47, 10. MR 3070913, https://doi.org/10.1214/ECP.v18-2824
  • [25] S. Sabanis, Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients, arXiv:1308.1796 (2014).
  • [26] Yi Shen, Qi Luo, and Xuerong Mao, The improved LaSalle-type theorems for stochastic functional differential equations, J. Math. Anal. Appl. 318 (2006), no. 1, 134-154. MR 2210878, https://doi.org/10.1016/j.jmaa.2005.05.026
  • [27] 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, https://doi.org/10.1137/120902318
  • [28] Fuke Wu, Xuerong Mao, and Lukas Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math. 115 (2010), no. 4, 681-697. MR 2658159, https://doi.org/10.1007/s00211-010-0294-7

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Additional Information

Łukasz Szpruch
Affiliation: School of Mathematics, The University of Edinburgh, EH9 3FD, Edinburgh, United Kingdom
Email: l.szpruch@ed.ac.uk

Xīlíng Zhāng
Affiliation: School of Mathematics, The University of Edinburgh, EH9 3FD, Edinburgh, United Kingdom
Email: xiling.zhang@ed.ac.uk

DOI: https://doi.org/10.1090/mcom/3219
Keywords: Stochastic differential equations, Lyapunov functions, Euler scheme, integrability, stability, comparison theorem
Received by editor(s): February 12, 2015
Received by editor(s) in revised form: January 4, 2016, May 23, 2016, and October 17, 2016
Published electronically: August 3, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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