How to simulate Lévy flights in a steep potential: An explicit splitting numerical scheme.

Area of Science:

• Numerical analysis
• Stochastic processes
• Computational physics

Background:

• Stochastic differential equations (SDEs) are crucial for modeling systems with inherent randomness.
• Heavy-tailed Lévy noise introduces complex dynamics not captured by standard Gaussian processes.
• Simulating SDEs with superlinear drift and Lévy noise poses significant numerical challenges, including potential solution explosion.

Purpose of the Study:

• To develop an effective explicit numerical scheme for simulating SDEs with specific challenging characteristics.
• To ensure the numerical scheme prevents solution explosion and accurately captures finite moments.
• To validate the scheme's performance in reproducing sub-Gaussian tail moments for Gaussian noise and its applicability to Lévy flights.

Main Methods:

• An explicit numerical scheme was designed for SDEs with confining superlinear drift.
• The scheme incorporates multiplicative heavy-tailed Lévy noise.
• The method focuses on stability and accurate moment estimation.

Main Results:

• The proposed scheme effectively prevents solution explosion.
• It accurately captures all finite moments of the solutions.
• In the Gaussian noise case, it correctly reproduces sub-Gaussian tail moments.

Conclusions:

• The numerical scheme provides a robust tool for simulating SDEs with superlinear drift and heavy-tailed Lévy noise.
• It is particularly suitable for approximating statistical moments of Lévy flights in steep potential landscapes.
• This method enhances the simulation accuracy for complex stochastic systems.