Boundary-preserving Lamperti-splitting schemes for some stochastic differential equations

Johan Ulander

doi:10.3934/jcd.2024015

Article Contents

2024, Volume 11, Issue 3: 289-317. Doi: 10.3934/jcd.2024015

This issue Previous Article Remarks on solitary waves in equations with nonlocal cubic terms Next Article The Prytz connections

Boundary-preserving Lamperti-splitting schemes for some stochastic differential equations

Johan Ulander

Chalmers University of Technology, Sweden

Received: August 2023

Revised: February 2024

Early access: March 11, 2024

Published online: March 11, 2024

The author is partially supported by the Swedish Research Council (VR) (projects nr. 2018 − 04443).

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Abstract

We propose and analyse boundary-preserving schemes for the strong approximations of some scalar SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The schemes consists of a Lamperti transform followed by a Lie–Trotter splitting. We prove $ L^{p}(\Omega) $-convergence of order 1, for every $ p \geq 1 $, of the schemes and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting schemes to other numerical schemes for SDEs.

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Mathematics Subject Classification: 60H10, 60H35, 65C30.

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Figure 1. Path comparison of the EM, SEM, TE and LS schemes applied to the SIS SDE with parameters $ \lambda = 4 $, $ x_{0} = 0.9 $, $ T = 0.4 $ and $ M = 50 $

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Figure 2. $ L^{2}(\Omega) $-errors on the interval $ [0,1] $ of the Lamperti-splitting scheme (LS) for the SIS SDE for different choices of $ \lambda>0 $ and reference lines with slopes $ 1/2 $ and 1. Averaged over 300 samples

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Figure 3. Path comparison of the EM, SEM, TEM and LS schemes applied to the Nagumo SDE with parameters $ \lambda = 4 $, $ x_{0} = 0.9 $, $ T = 0.4 $ and $ M = 50 $

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Figure 4. $ L^{2}(\Omega) $-errors on the interval $ [0,1] $ of the Lamperti-splitting scheme (LS) for the Nagumo SDE for different choices of $ \lambda>0 $ and reference lines with slopes $ 1/2 $ and 1. Averaged over 300 samples

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Figure 5. Path comparison of the EM, SEM, TE and LS schemes applied to the Allen–Cahn type SDE with parameters $ \lambda = 3 $, $ x_{0} = 0.9 $, $ T = 0.4 $ and $ M = 50 $

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Figure 6. $ L^{2}(\Omega) $-errors on the interval $ [0,1] $ of the Lamperti-splitting scheme (LS) for the Allen–Cahn type SDE for different choices of $ \lambda>0 $ and reference lines with slopes $ 1/2 $ and 1. Averaged over 300 samples

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Table 1. Proportion of samples containing only values in $ (0,1) $ out of 100 simulated sample paths for the Lamperti-splitting scheme (LS), the Euler–Maruyama scheme (EM), the semi-implicit Euler–Maruyama scheme (SEM), and the tamed Euler scheme (TE) for the SIS SDE for different choices of $ \lambda>0 $. The parameters used are: $ T = 1 $, $ \Delta t = 10^{-3} $ and with $ x_{0} $ uniformly distributed on $ (0,1) $ for each sample

$ \lambda $	LS	EM	SEM	TE
6	$ 100/100 $	$ 100/100 $	$ 100/100 $	$ 100/100 $
7	$ 100/100 $	$ 94/100 $	$ 89/100 $	$ 92/100 $
8	$ 100/100 $	$ 71/100 $	$ 63/100 $	$ 70/100 $

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Table 2. Proportion of samples containing only values in $ (0,1) $ out of 100 simulated sample paths for the Lamperti-splitting scheme (LS), the Euler–Maruyama scheme (EM), the semi-implicit Euler–Maruyama scheme (SEM), and the tamed Euler scheme (TE) for the Nagumo SDE for different choices of $ \lambda>0 $. The parameters used are: $ T = 1 $, $ \Delta t = 10^{-3} $ and with $ x_{0} $ uniformly distributed on $ (0,1) $ for each sample

$ \lambda $	LS	EM	SEM	TE
6	$ 100/100 $	$ 100/100 $	$ 100/100 $	$ 100/100 $
7	$ 100/100 $	$ 95/100 $	$ 97/100 $	$ 95/100 $
8	$ 100/100 $	$ 75/100 $	$ 77/100 $	$ 73/100 $

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Table 3. Proportion of samples containing only values in $ (-1,1) $ out of 100 simulated sample paths for the Lamperti-splitting scheme (LS), the Euler–Maruyama scheme (EM), the semi-implicit Euler–Maruyama scheme (SEM), and the tamed Euler scheme (TE) for the Allen–Cahn type SDE for different choices of $ \lambda>0 $, $ T = 1 $, $ \Delta t = 10^{-3} $ and with $ x_{0} $ uniformly distributed on $ (-1,1) $ for each sample

$ \lambda $	LS	EM	SEM	TE
3	$ 100/100 $	$ 100/100 $	$ 100/100 $	$ 100/100 $
3.3	$ 100/100 $	$ 97/100 $	$ 97/100 $	$ 95/100 $
3.6	$ 100/100 $	$ 74/100 $	$ 89/100 $	$ 82/100 $

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References

[1]	A. Alfonsi, Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process, Statist. Probab. Lett., 83 (2013), 602-607. doi: 10.1016/j.spl.2012.10.034.
[2]	S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.
[3]	C.-E. Bréhier, D. Cohen and J. Ulander, Analysis of a positivity-preserving splitting scheme for some nonlinear stochastic heat equations, preprint, 2023, arXiv: 2302.08858.
[4]	C.-E. Bréhier, D. Cohen and J. Ulander, Positivity-preserving Schemes for Some Nonlinear Stochastic PDEs, Proceedings of the Sixteenth International Conference Zaragoza-Pau on Mathematics and its Applications 2022.
[5]	C.-E. Bréhier and L. Goudenège, Analysis of some splitting schemes for the stochastic Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4169-4190. doi: 10.3934/dcdsb.2019077.
[6]	L. Chen, S. Gan and X. 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 pp. doi: 10.1016/j.cam.2021.113482.
[7]	R. Corless, G. Gonnet, D. Hare, J. Jeffrey and D. Knuth, On the LambertW function, Adv Comput Math, 5 (1996), 329-359. doi: 10.1007/BF02124750.
[8]	O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, Princeton, NJ, 2013.
[9]	T. Funaki, The scaling limit for a stochastic PDE and the separation of phases, Probability Theory and Related Fields, 102 (1995), 221-288. doi: 10.1007/BF01213390.
[10]	A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71 (2011), 876-902. doi: 10.1137/10081856X.
[11]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Reprint of the second (2006) edition, Springer, Heidelberg, 2010.
[12]	N. Halidias, Constructing positivity preserving numerical schemes for the two-factor CIR model, Monte Carlo Methods Appl., 21 (2015), 313-323. doi: 10.1515/mcma-2015-0109.
[13]	N. Halidias and I. S. Stamatiou, Boundary preserving explicit scheme for the Aït-Sahalia model, Discrete and Continuous Dynamical Systems - B, 28 (2023), 648-664. doi: 10.3934/dcdsb.2022092.
[14]	M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641. doi: 10.1214/11-AAP803.
[15]	S. Karlin and H. E. Taylor, A Second Course in Stochastic Processes, Academic Press, 1981.
[16]	C. Kelly and G. J. Lord, An adaptive splitting method for the Cox-Ingersoll-Ross process, Appl. Numer. Math., 186 (2023), 252-273. doi: 10.1016/j.apnum.2023.01.014.
[17]	Y. Kiouvrekis and I. S. Stamatiou, Domain preserving and strongly converging explicit scheme for the stochastic SIS epidemic model, preprint, 2023, arXiv: 2307.14404.
[18]	F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 3$^{rd}$ edition, Imperial Collage Press, 2012. doi: 10.1142/p821.
[19]	P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations, LMS J. Comput. Math., 10 (2007), 235-253. doi: 10.1112/S1461157000001388.
[20]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.
[21]	G. J. Lord, C. E. Powell and T. Shardlow, An Introduction to Computational Stochastic PDEs, Cambridge University Press, New York, 2014. doi: 10.1017/CBO9781139017329.
[22]	H. P. McKean Jr., Nagumo's equation, Advances in Mathematics, 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X.
[23]	R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434. doi: 10.1017/S0962492902000053.
[24]	J. K. Møller and H. Madsen, From State Dependent Diffusion to Constant Diffusion in Stochastic Differential Equations by the Lamperti Transform, Technical University of Denmark, DTU Informatics, Building 321. IMM-Technical Report-2010-16, 2010.
[25]	E. Moro and H. Schurz, Boundary preserving semianalytic numerical algorithms for stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 1525-1549. doi: 10.1137/05063725X.
[26]	A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs defined in a domain, Numer. Math., 128 (2014), 103-136. doi: 10.1007/s00211-014-0606-4.
[27]	S. Sabanis, Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients, Ann. Appl. Probab., 26 (2016), 2083-2105. doi: 10.1214/15-AAP1140.
[28]	H. Schurz, Numerical regularization for SDEs: Construction of nonnegative solutions, Dyn. Syst. Appl., 5 (1996), 323-351.
[29]	H. Yang and J. Huang, First order strong convergence of positivity preserving logarithmic Euler-Maruyama method for the stochastic SIS epidemic model, Appl. Math. Lett., 121 (2021), 7 pp. doi: 10.1016/j.aml.2021.107451.
[30]	H. Yang and J. Huang, Strong convergence and extinction of positivity preserving explicit scheme for the stochastic SIS epidemic model, Numer. Algor., 2023.
[31]	Y. Yi, Y. Hu and J. Zhao, Positivity preserving logarithmic Euler-Maruyama type scheme for stochastic differential equations, Commun. Nonlinear Sci. Numer. Simul., 101 (2021), 21 pp. doi: 10.1016/j.cnsns.2021.105895.
[32]	B. Øksendal, Stochastic Differential Equations, 6$^{th}$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.