On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions

M. Gerencser, A. Jentzen, D. Salimova, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473 (2017).


Journal Article | Published | English
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Abstract
In a recent article (Jentzen et al. 2016 Commun. Math. Sci. 14, 1477–1500 (doi:10.4310/CMS.2016.v14. n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d ? {4, 5, . . .}, there exist d-dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two (d = 2) and three (d = 3) space dimensions.
Publishing Year
Date Published
2017-11-01
Journal Title
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume
473
Issue
2207
Article Number
0104
ISSN
IST-REx-ID

Cite this

Gerencser M, Jentzen A, Salimova D. On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2017;473(2207). doi:10.1098/rspa.2017.0104
Gerencser, M., Jentzen, A., & Salimova, D. (2017). On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2207). https://doi.org/10.1098/rspa.2017.0104
Gerencser, Mate, Arnulf Jentzen, and Diyora Salimova. “On Stochastic Differential Equations with Arbitrarily Slow Convergence Rates for Strong Approximation in Two Space Dimensions.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2207 (2017). https://doi.org/10.1098/rspa.2017.0104.
M. Gerencser, A. Jentzen, and D. Salimova, “On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 473, no. 2207, 2017.
Gerencser M, Jentzen A, Salimova D. 2017. On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 473(2207).
Gerencser, Mate, et al. “On Stochastic Differential Equations with Arbitrarily Slow Convergence Rates for Strong Approximation in Two Space Dimensions.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 473, no. 2207, 0104, Royal Society of London, 2017, doi:10.1098/rspa.2017.0104.

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