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Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence

Published online by Cambridge University Press:  13 November 2007

Abdel Berkaoui ,
Mireille Bossy  and
Awa Diop
Abdel Berkaoui
Affiliation:
Dept of Statistics, University of Warwick, Gibbet Hill road, Coventry CV4 7AL, UK; berkaoui@stats.warwick.ac.uk
Mireille Bossy
Affiliation:
OMEGA project, INRIA Sophia Antipolis, 2004 route des Lucioles, B.P. 93, 06902 Sophia-Antipolis Cedex, France; mireille.bossy@sophia.inria.fr; ADiop@bbspinc.com
Awa Diop
Affiliation:
OMEGA project, INRIA Sophia Antipolis, 2004 route des Lucioles, B.P. 93, 06902 Sophia-Antipolis Cedex, France; mireille.bossy@sophia.inria.fr; ADiop@bbspinc.com
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Abstract

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 asymmetrized version of the Euler scheme. This symmetrized version iseasy to simulate on a computer. We prove its strong convergence and obtain the same rate ofconvergence as when the coefficients are Lipschitz.

Keywords

Discretization schemestrong convergenceCIR process.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 1 - 11
Copyright
© EDP Sciences, SMAI, 2008

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References

Alfonsi, A., On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11 (2005) 355–384. CrossRef
Berkaoui, A., Euler scheme for solutions of stochastic differential equations. Potugalia Mathematica Journal 61 (2004) 461–478.
M. Bossy and A. Diop, Euler scheme for one dimensional SDEs with a diffusion coefficient function of the form |x|α , a in [ 1/2,1). Annals Appl. Prob. (Submitted).
Bossy, M., Gobet, E. and Talay, D., A symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41 (2004) 877–889.
Cox, J., Ingersoll, J.E. and Ross, S.A., A theory of the term structure of the interest rates. Econometrica 53 (1985) 385–407. CrossRef
Deelstra, G. and Delbaen, F., Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stochastic Models Data Anal. 14 (1998) 77–84. 3.0.CO;2-2>CrossRef
O. Faure, Simulation du Mouvement Brownien et des Diffusions. Ph.D. thesis, École nationale des ponts et chaussées (1992).
P.S. Hagan, D. Kumar, A.S. Lesniewski and D.E. Woodward, Managing smile risk. WILMOTT Magazine (September, 2002).
Hull, J.C. and White, A., Pricing interest-rate derivative securities. Rev. Finan. Stud. 3 (1990) 573–592. CrossRef
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988).