Quasi-Sure Convergence Rate of Euler Scheme for Stochastic Differential Equations

Let Xt(x) be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt(x) be the Euler discretization scheme of SDEs with step 2−n. In this note, we prove that for any R > 0 and γ ∈ (0,1/2), supt∈[0,1],|x|≤R|Xtn(x,ω)−Xt(x,ω)|≤ξR,γ(ω)2−nγ,     n≥1,     q.e.,where ξR,γ(ω)ξR,γ(ω) is quasi-everywhere finite.