Approximation of SDEs: a stochastic sewing approach

Area of Science:

• Stochastic Analysis
• Numerical Analysis
• Probability Theory

Background:

• Stochastic differential equations (SDEs) are crucial for modeling complex systems.
• Approximation schemes like Euler-Maruyama are vital for solving SDEs.
• Understanding convergence rates is key to the reliability of these approximations.

Purpose of the Study:

• To develop a new framework for error analysis of SDE approximations.
• To leverage the stochastic sewing lemma for improved convergence rate analysis.
• To investigate the Euler-Maruyama scheme's performance under non-regular drift conditions.

Main Methods:

• Utilizing and extending the stochastic sewing lemma.
• Applying a novel error analysis approach to SDE approximations.
• Analyzing convergence rates for SDEs driven by fractional Brownian motion and standard Brownian noise.

Main Results:

• Demonstrated convergence of the Euler-Maruyama scheme for SDEs with non-regular drift driven by fractional Brownian motion.
• Established strong and almost sure convergence rates of $1/2$ for specific non-regular drift conditions.
• Derived an almost optimal convergence rate of $1/2$ for SDEs with multiplicative standard Brownian noise.

Conclusions:

• The stochastic sewing lemma provides a powerful tool for analyzing SDE approximations.
• The findings offer new insights into the convergence properties of numerical methods for SDEs.
• The established conditions for drift regularity are optimal, aligning with strong uniqueness requirements.