Information-based complexity
Error analysis of a randomized numerical method
Numerische Mathematik
The Quantum Setting with Randomized Queries for Continuous Problems
Quantum Information Processing
Almost optimal solution of initial-value problems by randomized and quantum algorithms
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Runge-Kutta methods for affinely controlled nonlinear systems
Journal of Computational and Applied Mathematics
The randomized complexity of initial value problems
Journal of Complexity
Optimal adaptive solution of initial-value problems with unknown singularities
Journal of Complexity
A random Euler scheme for Carathéodory differential equations
Journal of Computational and Applied Mathematics
Adaptive Itô-Taylor algorithm can optimally approximate the Itô integrals of singular functions
Journal of Computational and Applied Mathematics
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We investigate pointwise approximation of the solution of a scalar stochastic differential equation in case when drift coefficient is a Caratheodory mapping and diffusion coefficient is only piecewise Holder continuous with Holder exponent @r@?(0,1]. Since under imposed assumptions drift is only measurable with respect to the time variable, the classical Euler algorithm does not converge in general to the solution of such equation. We give a construction of the randomized Euler scheme and prove that it has the error O(n^-^m^i^n^{^@r^,^1^/^2^}), where n is the number of discretization points. We also investigate the optimality of the defined algorithm.