Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations

Area of Science:

• Numerical Analysis
• Stochastic Differential Equations
• Dynamical Systems

Background:

• Semi-linear stochastic delay differential equations (SLSDDEs) are crucial in modeling complex systems.
• Stability and convergence of numerical methods are essential for reliable simulations of SLSDDEs.
• Existing methods often rely on Lyapunov functions or implicit schemes for stability analysis.

Purpose of the Study:

• To investigate the strong convergence of the exponential Euler method for SLSDDEs.
• To analyze the mean-square exponential stability of the exponential Euler method for SLSDDEs.
• To demonstrate that the explicit exponential Euler method offers stability comparable to implicit methods.

Main Methods:

• Utilizing the definition of the logarithmic norm to analyze exponential stability in mean square.
• Proving the strong convergence order of the exponential Euler approximation.
• Developing an explicit numerical scheme for SLSDDEs.

Main Results:

• The exponential Euler approximation solution converges to the analytic solution with a strong order of convergence.
• The exponential Euler method demonstrates mean-square exponential stability for SLSDDEs.
• The explicit exponential Euler method achieves exponential stability in mean square for any step size.

Conclusions:

• The proposed explicit exponential Euler method is a viable and stable numerical approach for SLSDDEs.
• The findings extend the understanding of numerical stability for stochastic delay differential equations.
• This method provides a robust alternative for simulating systems described by SLSDDEs.