Numerical Generation of Trajectories Statistically Consistent with Stochastic Differential Equations

Area of Science:

• Computational Physics
• Numerical Analysis
• Stochastic Processes

Background:

• Stochastic Differential Equations (SDEs) model complex systems with inherent randomness.
• Accurate numerical methods are crucial for simulating SDEs, but often require direct noise realization.
• Existing methods like the Milstein algorithm have limitations in accuracy and computational efficiency.

Purpose of the Study:

• To develop a novel, weak second-order numerical method for SDE trajectory generation.
• To bypass direct noise realization for improved computational efficiency.
• To achieve high accuracy by reproducing cumulants of the state variable.

Main Methods:

• A weak second-order numerical scheme is proposed, updating system states with independent Gaussian random variables.
• The method reproduces the first three cumulants of the state variable to second order in time-step size.
• The update rule is derived from the Fokker-Planck equation in arbitrary dimensions.

Main Results:

• The developed method demonstrates high accuracy, outperforming the standard Milstein algorithm.
• Accuracy is validated using Büttiker's ratchet as a test case.
• The method's second-order accuracy in time-step size is confirmed.

Conclusions:

• The proposed numerical method offers an accurate and efficient alternative for SDE trajectory generation.
• It provides a foundation for extending to higher-order approximations of SDE solutions.
• This approach has broad applicability in fields relying on stochastic modeling.