A Milstein-based free knot spline approximation for stochastic differential equations

Abstract

This paper presents a pathwise approximation of scalar stochastic differential equations by polynomial splines with free knots. The pathwise distance between the solution and its approximation is measured globally on the unit interval in the L ∞ -norm, and the expectation of this distance is of concern here. We introduce an approximation method X ̂ k with k free knots which is based on asymptotic optimal approximation of a scalar Brownian motion by splines with free knots. For general stochastic differential equations we establish an upper bound of order 1 / k with an explicit asymptotic constant for the approximation error of X ̂ k . In the particular case of equations with additive noise this asymptotic upper bound is sharp.