The optimal free knot spline approximation of stochastic differential equations with additive noise

TL;DR

This paper investigates the best possible pathwise approximation of stochastic differential equations with additive noise using polynomial splines with free knots, establishing sharp bounds and an optimal approximation method.

Contribution

It provides sharp bounds for the minimal approximation error and introduces an optimal method combining Euler scheme and spline approximation of Brownian motion.

Findings

Sharp bounds for approximation error with free knots

Optimal approximation method achieves the best order

Combines Euler scheme with spline approximation of Brownian motion

Abstract

In this paper we analyse the pathwise approximation of 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 $L_{\infty}$-norm, and we study the expectation of this distance. For equations with additive noise we obtain sharp lower and upper bounds for the minimal error in the class of arbitrary spline approximation methods, which use $k$ free knots. The optimal order is achieved by an approximation method $\hat{X}_{k}^{\dagger}$, which combines an Euler scheme on a coarse grid with an optimal spline approximation of the Brownian motion $W$ with $k$ free knots.