Free-Knot Spline Approximation of Stochastic Processes

We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the $s$-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average $L_p$-distance to the splines spaces, as the (expected) number $k$ of free knots tends to infinity.Comment: 23 page

Non-uniform time discretization and lower bounds for approximation of stochastic heat equations

SIGLEAvailable from TIB Hannover: RR 4487(2004,18) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Best rates of convergence for strong approximation of SDE's at a single point

SIGLEAvailable from TIB Hannover: RR 4487(2003,26) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Spatial sampling design for prediction with estimated parameters

We study spatial sampling design for prediction of stationary isotropic Gaussian pro-cesses with estimated parameters of the covariance function. The key issue is how to incor-porate the parameter uncertainty into design criteria to correctly represent the uncertainty in prediction. Several possible design criteria are discussed that incorporate the parameter uncertainty. A simulated annealing algorithm is employed to search for the optimal design of small sample size and a two-step algorithm is proposed for moderately large sample sizes. Simulation results are presented for the Matérn class of covariance functions. An example of redesigning the air monitoring network in EPA Region 5 for monitoring sulfur dioxide is given to illustrate the possible differences our proposed design criterion can make in practice. Key Words: Fisher information matrix; Geostatistics; Kriging; Kullback divergence; Op-timization; Simulated annealing