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

Abdel Berkaoui; Mireille Bossy; Awa Diop

ESAIM: Probability and Statistics (2007)

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

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topBerkaoui, Abdel, Bossy, Mireille, and Diop, Awa. "Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence." ESAIM: Probability and Statistics 12 (2007): 1-11. <http://eudml.org/doc/104396>.

@article{Berkaoui2007,

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 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.; discretization scheme; stochastic differential equations; Euler scheme},

language = {eng},

month = {11},

pages = {1-11},

publisher = {EDP Sciences},

title = {Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence},

url = {http://eudml.org/doc/104396},

volume = {12},

year = {2007},

}

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

DA - 2007/11//

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|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.

LA - eng

KW - Discretization scheme; strong convergence; CIR process.; discretization scheme; stochastic differential equations; Euler scheme

UR - http://eudml.org/doc/104396

ER -

## References

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- A. Berkaoui, Euler scheme for solutions of stochastic differential equations. Potugalia Mathematica Journal61 (2004) 461–478. Zbl1065.60061
- 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).
- 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. Zbl1076.65009
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- 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. Zbl0915.60064
- 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).
- J.C. Hull and A. White, Pricing interest-rate derivative securities. Rev. Finan. Stud.3 (1990) 573–592.
- I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988). Zbl0638.60065

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