Mehdi Slassi
The uniform free-knot spline approximation of Stochastic Differential Equations
Abstract.
We analyze the pathwise approximation of scalar stochastic differential equations (SDE) 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 the expectation of this distance is of concern here. We introduce a numerical method \(\widehat{X}_{k}\) with \(k\) free knots which is based on asymptotic optimal approximation of a scalar Brownian motion by splines with free knots. For general SDEs, we establish an upper bound of order \(1/\sqrt{k}\) with an explicit asymptotic constant for the approximation error of \(\widehat{X}_{k}\). In particular case of SDEs with additive noise this asymptotic upper bound is sharp.
