Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion...

Area of Science:

• Numerical Analysis
• Stochastic Processes
• Computational Mathematics

Background:

• Stochastic differential equations (SDEs) are crucial for modeling complex systems in science and engineering.
• The Euler-Maruyama method is a standard numerical technique for approximating SDE solutions.
• Existing methods face challenges with SDEs featuring discontinuous drift or degenerate diffusion coefficients.

Purpose of the Study:

• To establish strong convergence rates for the Euler-Maruyama method under challenging SDE conditions.
• To provide a rigorous mathematical proof for the method's improved accuracy.
• To enhance the reliability of numerical solutions for complex stochastic models.

Main Methods:

• Development of a novel proof strategy based on error estimation.
• Comparison of the Euler-Maruyama scheme with a tailored numerical approach.
• Application of the Euler-Maruyama scheme to a transformed version of the original SDE.

Main Results:

• Proof of strong convergence of order [Formula: see text] for arbitrarily small [Formula: see text].
• Demonstration of the method's effectiveness even with discontinuous drift and degenerate diffusion.
• Quantification of the convergence rate under specific mathematical conditions.

Conclusions:

• The Euler-Maruyama method, with the proposed modifications and analysis, achieves significant convergence for complex SDEs.
• This work advances the numerical treatment of multidimensional SDEs with challenging coefficients.
• The findings are relevant for computational finance, physics, and engineering applications relying on SDEs.