Continuity modulus of stochastic homeomorphism flows for SDEs with non-Lipschitz coefficients

Let Xt (x) solve the following Itô-type SDE (denoted by EQ.(σ, b, x)) in Rd dXt = σ(Xt ) · dWt +b(Xt ) dt, X0 = x ∈ Rd . Assume that for anyN >0 and some CN > 0 b(x) −b(y) + ∇σ(x)−∇σ(y) CN|x −y| log |x − y|−1 ∨ 1 , |x|, |y| N, where ∇ denotes the gradient, and the explosion times of EQ.(σ, b, x) and EQ.(σ, tr(∇σ · σ) − b, x) are infinite for each x ∈ Rd . Then we prove that for fixed t > 0, x →X−1 t (x) is α(t)-order locally Hölder continuous a.s., where α(t) ∈ (0, 1) is exponentially decreasing to zero as the time goes to infinity. Moreover, for almost all ω, the inverse flow (t, x) →X−1 t (x,ω) is bicontinuous. © 2006 Elsevier Inc. All rights reserved.