Approximation and optimization on the Wiener space
Journal of Complexity
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
The optimal free knot spline approximation of stochastic differential equations with additive noise
Journal of Computational and Applied Mathematics
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This paper presents a pathwise approximation of scalar 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 the expectation of this distance is of concern here. We introduce an approximation method 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 stochastic differential equations we establish an upper bound of order 1/k with an explicit asymptotic constant for the approximation error of X@^"k. In the particular case of equations with additive noise this asymptotic upper bound is sharp.