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
Keywords: discretization scheme, strong convergence, CIR process 
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     title = {Euler scheme for {SDEs} with {non-Lipschitz} diffusion coefficient : strong convergence},
     journal = {ESAIM: Probability and Statistics},
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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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

[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). | Zbl | MR

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