Average n-widths of the Wiener space in the L∞ -norm
Journal of Complexity
Nonlinear approximation of random functions
SIAM Journal on Applied Mathematics
The optimal discretization of stochastic differential equations
Journal of Complexity
Relations between classical, average, and probabilistic Kolmogorov widths
Journal of Complexity
Information-based nonlinear approximation: an average case setting
Journal of Complexity
Free-knot spline approximation of stochastic processes
Journal of Complexity
A Milstein-based free knot spline approximation for stochastic differential equations
Journal of Complexity
On the importance of combining wavelet-based nonlinear approximation with coding strategies
IEEE Transactions on Information Theory
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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"~-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 X@^"k^@?, which combines an Euler scheme on a coarse grid with an optimal spline approximation of the Brownian motion W with k free knots.