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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.

##### MSC:
 65C30 Stochastic differential and integral equations 60H35 Computational methods for stochastic equations 60H10 Stochastic ordinary differential equations 60H35 Computational methods for stochastic equations 34F05 ODE with randomness 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L20 Stability and convergence of numerical methods for ODE
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