Our interest lies within the approximation of a stochastic process using functions from a nonlinear manifold, which may consist, e.g., of piecewise polynomial splines with a (random or fixed) number of (freely and path-dependently chosen) knots. To be more precise, we ask for the minimal error possible using at most n knots (on the average) and study the asymptotics as n tends to infinity.
Rough (and sometimes sharp) asymptotics were found for diffusion processes, stable Levy processes and fractional Brownian motion. For the case of a Brownian motion or a diffusion process, it was shown that there is at most an improvement in terms of logarithmic factors in the error rate compared to naive piecwiese linear equidistant interpolation; however, for more general (e.g., symmetric stable Levy) processes, it turned out that nonlinear approximation yields significantly better rates than linear approximation. In both cases, we aim at constructing simulation methods of such nonlinear approximations with a guaranteed error, and future work will include wavelet-based methods as well.

J. Creutzig, M. Lifshits (St. Petersburg University), W. Linde (Universität Jena), T. Müller-Gronbach (Universität Passau), L. Plaskota (University of Warsaw), K. Ritter

• J. Creutzig, T. Müller-Gronbach, K. Ritter, Free-knot spline approximation of stochastic processes, J. Complexity 23, 867-889, 2007. Preprint

Partially supported by the Deutsche Forschungsgemeinschaft, within a DFG-RFBR Grant on Geometry and Asymptotics of Random Structures