Weak convergence analysis of the symmetrized Euler scheme for one dimensional SDEs with diffusion coefficient |x|^a, a in [1/2,1)

2015/08/19 by Mireille Bossy, Bossy, Mireille, Awa Diop +1
Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1508.04573

openalex publication_date 2015/08/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper, we are interested in the time discrete approximation of Ef(X(T)) when X is the solution of a stochastic differential equation with a diffusion coefficient function of the form |x|^a. We propose a symmetrized version of the Euler scheme, applied to X. The symmetrized version is very easy to simulate on a computer. For smooth functions f, we prove the Feynman Kac representation u(t,x) = E_t,x f(X(T)), for u(t,x) solving the associated Kolmogorov PDE and we obtain the upper-bounds on the spatial derivatives of u up to the order four. Then we show that the weak error of our symmetrized scheme is of order one, as for the classical Euler scheme.

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