An approximation of a nonlinear integral equation driven by a function of bounded p-variation

The uniform distance between the solution of a nonlinear equation driven by a function h with bounded p-variation and its Milstein-type approximation is estimated by δn ∨ γp(n) ∨ γ p2 (n), where δn = max(tk - tk-1) is the maximum step size of the approximation on the interval [0, T], γp(n) = max vp1/p (h; [tk-1, tk]), 1 < p < 2, and vp(h; [tk-1, tk]) is the p-variation of the function h on [tk-1, tk]. In particular, if h is a Lipschitz function of order α, then the uniform distance has the bound δnα for δn < 1. © 1999 Kluwer Academic/Plenum Publishers.