Related Experiment Video
Updated: Feb 17, 2026
06:44
Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
602
On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space
Máté Gerencsér1, Arnulf Jentzen2, Diyora Salimova2
1Institute of Science and Technology Austria, Klosterneuburg, Austria.
Proceedings. Mathematical, Physical, and Engineering Sciences
|December 12, 2017
Summary
Researchers found that approximation methods for stochastic differential equations can exhibit slow convergence, even in lower dimensions. This slow convergence phenomenon is proven to occur in 2 and 3 dimensions, impacting computational mathematics.
Area of Science:
- Mathematics
- Computational Science
- Stochastic Analysis
Background:
- Previous work established slow convergence for d-dimensional stochastic differential equations (SDEs) where d>=4.
- This phenomenon was demonstrated for SDEs with infinitely often differentiable and globally bounded coefficients.
- Approximation methods based on finite observations of Brownian motion were shown to have limitations.
Purpose of the Study:
- To investigate whether the slow convergence phenomenon observed in higher dimensions also occurs in lower dimensions (d=2 and d=3).
- To extend the findings on the limitations of approximation methods for SDEs to two and three space dimensions.
Main Methods:
- The study builds upon the theoretical framework established in Jentzen et al. (2016).
- It involves the construction and analysis of specific d-dimensional stochastic differential equations.
- Mathematical proofs are used to demonstrate the convergence rates of approximation methods.
Main Results:
- The paper proves that the slow convergence phenomenon, previously shown for d>=4, also arises in two (d=2) and three (d=3) space dimensions.
- This indicates that for certain SDEs, approximation methods cannot converge faster than an arbitrarily slow specified speed, even in low dimensions.
- The coefficients of these SDEs are infinitely often differentiable and globally bounded.
Conclusions:
- The limitations on the convergence speed of approximation methods for SDEs are not confined to high dimensions.
- This finding has significant implications for numerical analysis and the practical computation of solutions to stochastic differential equations.
- Further research may explore the precise nature of these limitations across various classes of SDEs.
Related Concept Videos
01:21Linear Approximation in Time Domain
379
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
379
01:06Divergence and Stokes' Theorems
3.8K
The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
3.8K
01:21Convergence of Fourier Series
440
The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
440
01:19Limits with Oscillating Discontinuities
494
An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the...
494
01:29Improper Integrals: Infinite Intervals
122
An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
122
01:27Linear Differential Equations
111
The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law...
111