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Euler scheme for SDEs with non-lipschitz diffusion coefficient: Strong convergence. (English) Zbl 1183.65004
Summary: We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form $\vert x\vert^\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.

65C30Stochastic differential and integral equations
60H35Computational methods for stochastic equations
60H10Stochastic ordinary differential equations
60H35Computational methods for stochastic equations
34F05ODE with randomness
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE
Full Text: DOI EuDML
[1] A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11 (2005) 355-384. Zbl1100.65007 MR2186814 · Zbl 1100.65007 · doi:10.1515/156939605777438569
[2] A. Berkaoui, Euler scheme for solutions of stochastic differential equations. Potugalia Mathematica Journal 61 (2004) 461-478. Zbl1065.60061 MR2113559 · Zbl 1065.60061 · eudml:51417
[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. Zbl1076.65009 MR2074829 · Zbl 1076.65009 · doi:10.1239/jap/1091543431
[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. MR785475 · Zbl 1274.91447
[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 MR1641781 · Zbl 0915.60064 · doi:10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2
[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 MR917065 · Zbl 0638.60065