Rare event simulation for steady-state probabilities via recurrency cycles.

Area of Science:

• Computational Mathematics
• Stochastic Processes
• Numerical Analysis

Background:

• Estimating rare event probabilities in continuous-state Markov chains is computationally challenging.
• Existing methods often lack efficiency for complex systems.
• Markov chains are fundamental models in various scientific domains.

Purpose of the Study:

• To develop a new, efficient algorithm for rare event probability estimation.
• To leverage the recurrent structure of Markov chains for improved computation.
• To address limitations of existing computational methods.

Main Methods:

• Development of the Recurrent Multilevel Splitting (RMS) algorithm.
• Utilizing the underlying recurrent structure of Markov chains.
• Integration with the Multilevel Splitting (MS) technique.
• Application to discrete-time R^d-valued Markov chains.

Main Results:

• The proposed RMS algorithm demonstrates significant computational efficiency gains.
• RMS outperforms standard Monte Carlo methods by several orders of magnitude.
• Validation through extensive simulation experiments, including complex nonlinear stochastic models.

Conclusions:

• RMS offers a computationally efficient approach for rare event probability estimation.
• The algorithm's effectiveness is demonstrated on challenging stochastic models.
• RMS provides a valuable tool for analyzing complex systems in fields like climate modeling.