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

Abdel Berkaoui; Mireille Bossy; Awa Diop

ESAIM: Probability and Statistics (2008)

  • Volume: 12, page 1-11
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Berkaoui, Abdel, Bossy, Mireille, and Diop, Awa. "Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence." ESAIM: Probability and Statistics 12 (2008): 1-11. <http://eudml.org/doc/246056>.

@article{Berkaoui2008,
abstract = {We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form $|x|^\alpha $, $\alpha \in [1/2,1)$. 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.},
author = {Berkaoui, Abdel, Bossy, Mireille, Diop, Awa},
journal = {ESAIM: Probability and Statistics},
keywords = {discretization scheme; strong convergence; CIR process; stochastic differential equations; Euler scheme},
language = {eng},
pages = {1-11},
publisher = {EDP-Sciences},
title = {Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence},
url = {http://eudml.org/doc/246056},
volume = {12},
year = {2008},
}

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
PB - EDP-Sciences
VL - 12
SP - 1
EP - 11
AB - We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form $|x|^\alpha $, $\alpha \in [1/2,1)$. 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.
LA - eng
KW - discretization scheme; strong convergence; CIR process; stochastic differential equations; Euler scheme
UR - http://eudml.org/doc/246056
ER -

References

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  2. [2] A. Berkaoui, Euler scheme for solutions of stochastic differential equations. Potugalia Mathematica Journal 61 (2004) 461–478. Zbl1065.60061MR2113559
  3. [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. [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. Zbl1076.65009MR2074829
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  6. [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. Zbl0915.60064MR1641781
  7. [7] O. Faure, Simulation du Mouvement Brownien et des Diffusions. Ph.D. thesis, École nationale des ponts et chaussées (1992). 
  8. [8] P.S. Hagan, D. Kumar, A.S. Lesniewski and D.E. Woodward, Managing smile risk. WILMOTT Magazine (September, 2002). 
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  10. [10] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988). Zbl0638.60065MR917065

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