A strong order 1.5 boundary preserving discretization scheme for scalar SDEs defined in a domain

Liu, Ruishu; Neuenkirch, Andreas; Wang, Xiaojie

doi:10.1090/mcom/4014

A strong order $1.5$ boundary preserving discretization scheme for scalar SDEs defined in a domain
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by Ruishu Liu, Andreas Neuenkirch and Xiaojie Wang;

Math. Comp. 94 (2025), 1815-1862

DOI: https://doi.org/10.1090/mcom/4014

Published electronically: December 27, 2024

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Abstract:

In this paper, we study the strong approximation of scalar stochastic differential equations (SDEs), which take values in a domain and have non-Lipschitz coefficients. By combining a Lamperti-type transformation with a semi-implicit discretization approach and a taming strategy, we construct a domain-preserving scheme that strongly converges under weak assumptions. Moreover, we show that this scheme has strong convergence order $1.5$ under additional assumptions on the coefficients of the SDE. In our scheme, the domain preservation is a consequence of the semi-implicit discretization approach, while the taming strategy allows controlling terms of the scheme that admit singularities but are required to obtain the desired order.

Our general convergence results are applied to various SDEs from applications, with sub-linearly or super-linearly growing and non-globally Lipschitz coefficients. Numerical experiments are presented to illustrate our theoretical findings.

References

Aurélien Alfonsi, Strong order one convergence of a drift implicit Euler scheme: application to the CIR process, Statist. Probab. Lett. 83 (2013), no. 2, 602–607. MR 3006996, DOI 10.1016/j.spl.2012.10.034
Adam Andersson and Raphael Kruse, Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition, BIT 57 (2017), no. 1, 21–53. MR 3608309, DOI 10.1007/s10543-016-0624-y
Abdel Berkaoui, Mireille Bossy, and Awa Diop, Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence, ESAIM Probab. Stat. 12 (2008), 1–11. MR 2367990, DOI 10.1051/ps:2007030
Wolf-Jürgen Beyn, Elena Isaak, and Raphael Kruse, Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes, J. Sci. Comput. 67 (2016), no. 3, 955–987. MR 3493491, DOI 10.1007/s10915-015-0114-4
Wolf-Jürgen Beyn, Elena Isaak, and Raphael Kruse, Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, J. Sci. Comput. 70 (2017), no. 3, 1042–1077. MR 3608332, DOI 10.1007/s10915-016-0290-x
Charles-Edouard Bréhier, Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme, ESAIM Math. Model. Numer. Anal. 56 (2022), no. 1, 151–175. MR 4376272, DOI 10.1051/m2an/2021089
Yongmei Cai, Junhao Hu, and Xuerong Mao, Positivity and boundedness preserving numerical scheme for the stochastic epidemic model with square-root diffusion term, Appl. Numer. Math. 182 (2022), 100–116. MR 4469088, DOI 10.1016/j.apnum.2022.07.019
K. C. Chan, G. A. Karolyi, F. A. Longstaff, and A. B. Sanders, An empirical comparison of alternative models of the short-term intrest rate, The Journal of Finance 47 (1992), no. 3, 1209–1227.
Jean-François Chassagneux, Antoine Jacquier, and Ivo Mihaylov, An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients, SIAM J. Financial Math. 7 (2016), no. 1, 993–1021. MR 3582409, DOI 10.1137/15M1017788
Lin Chen, Siqing Gan, and Xiaojie Wang, First order strong convergence of an explicit scheme for the stochastic SIS epidemic model, J. Comput. Appl. Math. 392 (2021), Paper No. 113482, 16. MR 4220738, DOI 10.1016/j.cam.2021.113482
John C Cox, The constant elasticity of variance option pricing model, Journal of Portfolio Management 23 (1996), no. 5, 15–17.
John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), no. 2, 385–407. MR 785475, DOI 10.2307/1911242
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
Gabriel G. Drimus, Options on realized variance by transform methods: a non-affine stochastic volatility model, Quant. Finance 12 (2012), no. 11, 1679–1694. MR 2984555, DOI 10.1080/14697688.2011.565789
Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085, DOI 10.1002/9780470316658
William Feller, Two singular diffusion problems, Ann. of Math. (2) 54 (1951), 173–182. MR 54814, DOI 10.2307/1969318
A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011), no. 3, 876–902. MR 2821582, DOI 10.1137/10081856X
Steven L Heston, A simple new formula for options with stochastic volatility, Course notes of Washington University in St. Louis, Missouri, 1997.
Desmond J. Higham and Peter E. Kloeden, An introduction to the numerical simulation of stochastic differential equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, [2021] ©2021. MR 4241457, DOI 10.1137/1.9781611976434
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
Sam Howison and Daniel Schwarz, Risk-neutral pricing of financial instruments in emission markets: a structural approach, SIAM J. Financial Math. 3 (2012), no. 1, 709–739. MR 3022174, DOI 10.1137/100815219
T. R. Hurd and A. Kuznetsov, Explicit formulas for Laplace transforms of stochastic integrals, Markov Process. Related Fields 14 (2008), no. 2, 277–290. MR 2437532
Martin Hutzenthaler and Arnulf Jentzen, 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. MR 4079431, DOI 10.1214/19-AOP1345
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
A. Jentzen, P. E. Kloeden, and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients, Numer. Math. 112 (2009), no. 1, 41–64. MR 2481529, DOI 10.1007/s00211-008-0200-8
Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
Samuel Karlin and Howard M. Taylor, A second course in stochastic processes, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 611513
P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations, LMS J. Comput. Math. 10 (2007), 235–253. MR 2320830, DOI 10.1112/S1461157000001388
Kristian Stegenborg Larsen and Michael Sørensen, Diffusion models for exchange rates in a target zone, Math. Finance 17 (2007), no. 2, 285–306. MR 2301773, DOI 10.1111/j.1467-9965.2006.00304.x
Ruishu Liu and Xiaojie Wang, A higher order positivity preserving scheme for the strong approximations of a stochastic epidemic model, Commun. Nonlinear Sci. Numer. Simul. 124 (2023), Paper No. 107258, 23. MR 4597411, DOI 10.1016/j.cnsns.2023.107258
Xuerong Mao, The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370–384. MR 3370415, DOI 10.1016/j.cam.2015.06.002
Xuerong Mao, Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 296 (2016), 362–375. MR 3430145, DOI 10.1016/j.cam.2015.09.035
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
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, DOI 10.1007/978-3-662-12616-5
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
Sotirios Sabanis and Ying Zhang, On explicit order 1.5 approximations with varying coefficients: the case of super-linear diffusion coefficients, J. Complexity 50 (2019), 84–115. MR 3907365, DOI 10.1016/j.jco.2018.09.004
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
A. W. van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3, Cambridge University Press, Cambridge, 1998. MR 1652247, DOI 10.1017/CBO9780511802256
Mark Veraar, The stochastic Fubini theorem revisited, Stochastics 84 (2012), no. 4, 543–551. MR 2966093, DOI 10.1080/17442508.2011.618883
Xiaojie Wang, Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients, Adv. Comput. Math. 49 (2023), no. 3, Paper No. 37, 48. MR 4601143, DOI 10.1007/s10444-023-10034-2
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
Xiaojie Wang, Jiayi Wu, and Bozhang Dong, Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition, BIT 60 (2020), no. 3, 759–790. MR 4132905, DOI 10.1007/s10543-019-00793-0
Hongfu Yang and Jianhua Huang, First order strong convergence of positivity preserving logarithmic Euler-Maruyama method for the stochastic SIS epidemic model, Appl. Math. Lett. 121 (2021), Paper No. 107451, 7. MR 4274899, DOI 10.1016/j.aml.2021.107451
Hongfu Yang and Jianhua Huang, Strong convergence and extinction of positivity preserving explicit scheme for the stochastic SIS epidemic model, Numer. Algorithms 95 (2024), no. 4, 1475–1502. MR 4719422, DOI 10.1007/s11075-023-01617-7
Yulian Yi, Yaozhong Hu, and Jingjun Zhao, Positivity preserving logarithmic Euler-Maruyama type scheme for stochastic differential equations, Commun. Nonlinear Sci. Numer. Simul. 101 (2021), Paper No. 105895, 21. MR 4268206, DOI 10.1016/j.cnsns.2021.105895
Jianwei Zhu, Modular pricing of options: an application of fourier analysis, Springer, Berlin, 2013.

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A strong order $1.5$ boundary preserving discretization scheme for scalar SDEs defined in a domain
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Abstract: