Optimal Error Rates for Strong Approximation of SDEs with Fractional Sobolev Drift Coefficient

Core Concepts

The L2-error rate (1 + s)/2 is essentially optimal for approximating SDEs with a drift coefficient of fractional Sobolev regularity.

Abstract

The article discusses the strong approximation of stochastic differential equations (SDEs) with bounded drift coefficients and Sobolev regularity. It proves that the equidistant Euler approximation achieves an L2-error rate of at least (1 + s)/2. The study focuses on cases where the drift coefficient has fractional Sobolev regularity between 1/2 and 1. The analysis employs techniques like coupling of noise to establish lower error bounds. The results suggest that no numerical method can improve the L2-error rate beyond (1 + s)/2 for bounded drift coefficients in W s,p spaces.

Stats

For every n ∈ N, there exists c ∈ (0, ∞) such that for all n ∈ N, E sup t∈[0,1] |Xt - XEn,t|pi1/p ≤ c n(1+s)/2-ε . Recently shown that for SDEs with piecewise differentiable drift coefficients and discontinuities, the best possible Lp-error rate achievable is at most 3/4 for all p ∈ [1, ∞). The Euler approximation XEn,1 achieves an Lp-error rate of at least 1/2 even if the drift coefficient µ is only bounded and measurable.

Quotes

"We employ the coupling of noise technique to bound the L2-error of an arbitrary approximation." "The estimate naturally leads to questioning whether better Lp-error rates can be achieved." "Results show that adaptive approximations may outperform non-adaptive methods under certain conditions."

Key Insights Distilled From

On optimal error rates for strong approximation of SDEs with a drift coefficient of fractional Sobolev regularity

by Simo... at arxiv.org 03-14-2024

https://arxiv.org/pdf/2402.13732.pdf

Deeper Inquiries

How does the study impact advancements in numerical methods for approximating SDEs

The study on optimal error rates for strong approximation of SDEs with a drift coefficient of fractional Sobolev regularity contributes significantly to advancements in numerical methods for approximating stochastic differential equations (SDEs). By analyzing the equidistant Euler approximation and proving lower error bounds, the research provides valuable insights into the performance of numerical schemes in capturing the behavior of SDE solutions accurately. This understanding is crucial for developing more efficient and reliable computational algorithms for solving complex stochastic systems.

What are potential counterarguments against the optimality of the L2-error rate proposed by the author

Potential counterarguments against the optimality of the L2-error rate proposed by the author could include considerations about specific cases or scenarios where alternative methods may outperform or provide better accuracy than the equidistant Euler approximation. These counterarguments might involve discussing limitations or assumptions made in the study, such as restrictions on certain types of drift coefficients or noise processes that could affect the generalizability of the results. Additionally, practical challenges in implementing numerical methods based on finitely many evaluations of driving Brownian motion could be raised as potential factors influencing error rates.

How does fooling algorithms relate to information-based complexity in this context

In this context, fooling algorithms are closely related to information-based complexity by providing a framework for constructing instances where admissible algorithms cannot distinguish between two problems based on available information. The concept aligns with ideas from information-based complexity theory, which focuses on understanding algorithmic efficiency and computational complexity under various constraints. By using fooling algorithms to analyze L2-approximation errors based on different discretizations and coupled driving Brownian motions, researchers can gain insights into how well numerical methods perform in approximating SDE solutions while considering information constraints and computational resources.