Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 1-11.

We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form |x|α, α∈[1/2,1). In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.

DOI : 10.1051/ps:2007030
Classification : 65C30, 60H35, 65C20
Mots-clés : discretization scheme, strong convergence, CIR process 
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     title = {Euler scheme for {SDEs} with {non-Lipschitz} diffusion coefficient : strong convergence},
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Berkaoui, Abdel; Bossy, Mireille; Diop, Awa. Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 1-11. doi : 10.1051/ps:2007030. https://numdam.org/articles/10.1051/ps:2007030/

[1] A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11 (2005) 355-384. | MR | Zbl

[2] A. Berkaoui, Euler scheme for solutions of stochastic differential equations. Potugalia Mathematica Journal 61 (2004) 461-478. | MR | Zbl

[3] M. Bossy and A. Diop, Euler scheme for one dimensional SDEs with a diffusion coefficient function of the form |x|a, a in [1/2,1). Annals Appl. Prob. (Submitted).

[4] M. Bossy, E. Gobet and D. Talay, A symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41 (2004) 877-889. | MR | Zbl

[5] J. Cox, J.E. Ingersoll and S.A. Ross, A theory of the term structure of the interest rates. Econometrica 53 (1985) 385-407. | MR

[6] G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stochastic Models Data Anal. 14 (1998) 77-84. | MR | Zbl

[7] O. Faure, Simulation du Mouvement Brownien et des Diffusions. Ph.D. thesis, École nationale des ponts et chaussées (1992).

[8] P.S. Hagan, D. Kumar, A.S. Lesniewski and D.E. Woodward, Managing smile risk. WILMOTT Magazine (September, 2002).

[9] J.C. Hull and A. White, Pricing interest-rate derivative securities. Rev. Finan. Stud. 3 (1990) 573-592.

[10] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988). | MR | Zbl

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  • Gottwald, Georg A.; Melbourne, Ian Simulation of Non-Lipschitz Stochastic Differential Equations Driven by α-Stable Noise: A Method Based on Deterministic Homogenization, Multiscale Modeling Simulation, Volume 19 (2021) no. 2, p. 665 | DOI:10.1137/20m1333183
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  • Liang, Xingyin; Sun, Youfa; Yao, Yuhang Key Technique of Almost Exact Simulation for Non-affine Heston Model, Journal of Physics: Conference Series, Volume 1624 (2020) no. 2, p. 022016 | DOI:10.1088/1742-6596/1624/2/022016
  • Ben Alaya, Mohamed; Kebaier, Ahmed; Tran, Ngoc Khue Local asymptotic properties for Cox‐Ingersoll‐Ross process with discrete observations, Scandinavian Journal of Statistics, Volume 47 (2020) no. 4, p. 1401 | DOI:10.1111/sjos.12494
  • Bourza, Mohamed; Benabdallah, Mohsine Convergence rate of Euler scheme for time-inhomogeneous SDEs involving the local time of the unknown process, Stochastic Models, Volume 36 (2020) no. 3, p. 452 | DOI:10.1080/15326349.2020.1748506
  • Hong, Jialin; Huang, Chuying; Kamrani, Minoo; Wang, Xu Optimal strong convergence rate of a backward Euler type scheme for the Cox–Ingersoll–Ross model driven by fractional Brownian motion, Stochastic Processes and their Applications, Volume 130 (2020) no. 5, p. 2675 | DOI:10.1016/j.spa.2019.07.014
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  • Szavits-Nossan, J; Evans, M R Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges, Journal of Statistical Mechanics: Theory and Experiment, Volume 2015 (2015) no. 12, p. P12008 | DOI:10.1088/1742-5468/2015/12/p12008
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  • Mao, Xuerong; Szpruch, Lukasz Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, Journal of Computational and Applied Mathematics, Volume 238 (2013), p. 14 | DOI:10.1016/j.cam.2012.08.015
  • Chen, Nan; Huang, Zhengyu Localization and Exact Simulation of Brownian Motion-Driven Stochastic Differential Equations, Mathematics of Operations Research, Volume 38 (2013) no. 3, p. 591 | DOI:10.1287/moor.2013.0585
  • Tse, S. T.; Wan, Justin W. L. Low-bias simulation scheme for the Heston model by Inverse Gaussian approximation, Quantitative Finance, Volume 13 (2013) no. 6, p. 919 | DOI:10.1080/14697688.2012.696678
  • Alfonsi, Aurélien Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process, Statistics Probability Letters, Volume 83 (2013) no. 2, p. 602 | DOI:10.1016/j.spl.2012.10.034
  • Mao, Xuerong; Szpruch, Lukasz Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics, Volume 85 (2013) no. 1, p. 144 | DOI:10.1080/17442508.2011.651213
  • Dangerfield, C. E.; Kay, D.; MacNamara, S.; Burrage, K. A boundary preserving numerical algorithm for the Wright-Fisher model with mutation, BIT Numerical Mathematics, Volume 52 (2012) no. 2, p. 283 | DOI:10.1007/s10543-011-0351-3
  • Dereich, Steffen; Neuenkirch, Andreas; Szpruch, Lukasz An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 468 (2012) no. 2140, p. 1105 | DOI:10.1098/rspa.2011.0505
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  • Pagès, Gilles; Panloup, Fabien Ergodic approximation of the distribution of a stationary diffusion: Rate of convergence, The Annals of Applied Probability, Volume 22 (2012) no. 3 | DOI:10.1214/11-aap779
  • Szpruch, Lukasz; Mao, Xuerong; Higham, Desmond J.; Pan, Jiazhu Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT Numerical Mathematics, Volume 51 (2011) no. 2, p. 405 | DOI:10.1007/s10543-010-0288-y
  • Glasserman, Paul; Kim, Kyoung-Kuk Gamma expansion of the Heston stochastic volatility model, Finance and Stochastics, Volume 15 (2011) no. 2, p. 267 | DOI:10.1007/s00780-009-0115-y
  • Dana, Saswati; Raha, Soumyendu Physically consistent simulation of mesoscale chemical kinetics: The non-negative FIS-α method, Journal of Computational Physics, Volume 230 (2011) no. 24, p. 8813 | DOI:10.1016/j.jcp.2011.07.032
  • Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E. Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 467 (2011) no. 2130, p. 1563 | DOI:10.1098/rspa.2010.0348
  • Mishura, Yuliya S.; Posashkova, Svitlana V. The rate of convergence of the Euler scheme to the solution of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion, Random Operators and Stochastic Equations, Volume 19 (2011) no. 1 | DOI:10.1515/rose.2011.003
  • Sbai, Mohamed; Jourdain, Benjamin High Order Discretization Schemes for Stochastic Volatility Models, SSRN Electronic Journal (2011) | DOI:10.2139/ssrn.1452727
  • Gyöngy, István; Rásonyi, Miklós A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients, Stochastic Processes and their Applications, Volume 121 (2011) no. 10, p. 2189 | DOI:10.1016/j.spa.2011.06.008
  • De Marco, Stefano Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions, The Annals of Applied Probability, Volume 21 (2011) no. 4 | DOI:10.1214/10-aap717
  • Zubchenko, V. P.; Mishura, Yu. S. Rate of convergence in the Euler scheme for stochastic differential equations with non-Lipschitz diffusion and Poisson measure, Ukrainian Mathematical Journal, Volume 63 (2011) no. 1, p. 49 | DOI:10.1007/s11253-011-0487-y
  • Lord, Roger; Koekkoek, Remmert; Dijk, Dick Van A comparison of biased simulation schemes for stochastic volatility models, Quantitative Finance, Volume 10 (2010) no. 2, p. 177 | DOI:10.1080/14697680802392496
  • Tse, Shu Tong; Wan, Justin W. L. Low-Bias Simulation Scheme for the Heston Model by Inverse Gaussian Approximation, SSRN Electronic Journal (2010) | DOI:10.2139/ssrn.1644977
  • Gauthier, Pierre; Possamai, Dylan Efficient Simulation of the Double Heston Model, SSRN Electronic Journal (2010) | DOI:10.2139/ssrn.1434853
  • Labbé, Chantal; Remillard, Bruno; Renaud, Jean-Francois A Simple Discretization Scheme for Nonnegative Diffusion Processes, with Applications to Option Pricing, SSRN Electronic Journal (2010) | DOI:10.2139/ssrn.1619989
  • Pagès, Gilles; Panloup, Fabien Approximation of the distribution of a stationary Markov process with application to option pricing, Bernoulli, Volume 15 (2009) no. 1 | DOI:10.3150/08-bej142
  • Platen, Eckhard; Rendek, Renatak Exact Scenario Simulation for Selected Multi-Dimensional Stochastic Processes, SSRN Electronic Journal (2009) | DOI:10.2139/ssrn.2172581
  • Glasserman, Paul; Kim, Kyoung-Kuk Gamma Expansion of the Heston Stochastic Volatility Model, SSRN Electronic Journal (2008) | DOI:10.2139/ssrn.1279850
  • Lord, Roger; Koekkoek, Remmert; van Dijk, Dick J. C. A Comparison of Biased Simulation Schemes for Stochastic Volatility Models, SSRN Electronic Journal (2008) | DOI:10.2139/ssrn.903116

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