Generating discrete-time constrained random walks and Lévy flights

Area of Science:

• Probability Theory
• Stochastic Processes
• Mathematical Physics

Background:

• Random walks are fundamental models in statistical physics and probability.
• Generating constrained trajectories, like bridges, is crucial for simulating complex systems.
• Existing methods may lack efficiency or generality for arbitrary jump distributions.

Purpose of the Study:

• To introduce an exact and efficient method for generating bridge trajectories of discrete-time random walks.
• To handle arbitrary jump distributions within the random walk model.
• To extend the method to other constrained random walk types.

Main Methods:

• Development of an effective jump distribution that incorporates the bridge constraint.
• Application of the method to discrete-time random walks with diverse jump distributions.
• Demonstration of computational efficiency and accuracy.

Main Results:

• The proposed method accurately generates bridge trajectories for discrete-time random walks.
• The method is shown to be highly efficient across various jump distributions.
• Successful generalization to other constrained random walks, including generalized bridges, excursions, and meanders.

Conclusions:

• The introduced method provides an exact and efficient solution for generating random walk bridge trajectories.
• This technique offers a powerful tool for analyzing constrained stochastic processes.
• The generalization expands its applicability to a broader range of path-dependent phenomena.