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

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 (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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  1. A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl.11 (2005) 355–384.  Zbl1100.65007
  2. A. Berkaoui, Euler scheme for solutions of stochastic differential equations. Potugalia Mathematica Journal61 (2004) 461–478.  Zbl1065.60061
  3. 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).  
  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.  Zbl1076.65009
  5. J. Cox, J.E. Ingersoll and S.A. Ross, A theory of the term structure of the interest rates. Econometrica53 (1985) 385–407.  Zbl1274.91447
  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.  Zbl0915.60064
  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).  Zbl0638.60065

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