Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients

We study the approximation of stochastic differential equations on domains. For this, we introduce modified ItA '-Taylor schemes, which preserve approximately the boundary domain of the equation under consideration. Assuming the existence of a unique non-exploding solution, we show that the modified ItA '-Taylor scheme of order gamma has pathwise convergence order gamma - epsilon for arbitrary epsilon > 0 as long as the coefficients of the equation are sufficiently differentiable. In particular, no global Lipschitz conditions for the coefficients and their derivatives are required. This applies for example to the so called square root diffusions.