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 
@article{PS_2008__12__1_0,
     author = {Berkaoui, Abdel and Bossy, Mireille and Diop, Awa},
     title = {Euler scheme for {SDEs} with {non-Lipschitz} diffusion coefficient : strong convergence},
     journal = {ESAIM: Probability and Statistics},
     pages = {1--11},
     publisher = {EDP-Sciences},
     volume = {12},
     year = {2008},
     doi = {10.1051/ps:2007030},
     mrnumber = {2367990},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2007030/}
}
TY  - JOUR
AU  - Berkaoui, Abdel
AU  - Bossy, Mireille
AU  - Diop, Awa
TI  - Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence
JO  - ESAIM: Probability and Statistics
PY  - 2008
SP  - 1
EP  - 11
VL  - 12
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2007030/
DO  - 10.1051/ps:2007030
LA  - en
ID  - PS_2008__12__1_0
ER  - 
%0 Journal Article
%A Berkaoui, Abdel
%A Bossy, Mireille
%A Diop, Awa
%T Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence
%J ESAIM: Probability and Statistics
%D 2008
%P 1-11
%V 12
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2007030/
%R 10.1051/ps:2007030
%G en
%F PS_2008__12__1_0
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. http://www.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

Cité par Sources :