Exact simulation of max-stable processes.

Area of Science:

• Extreme Value Theory
• Spatial Statistics
• Stochastic Processes

Background:

• Max-stable processes are crucial for modeling spatial extreme events.
• Existing simulation algorithms are often inexact and computationally intensive.
• The complex structure of max-stable processes hinders direct simulation.

Purpose of the Study:

• To develop a new algorithm for the exact simulation of max-stable processes at finite locations.
• To improve the computational efficiency of simulating spatial extreme events.
• To generalize existing simulation methods for Brown-Resnick processes.

Main Methods:

• Development of an exact simulation algorithm based on identifying and simulating extremal functions.
• Generalization of the Dieker & Mikosch (2015) algorithm for Brown-Resnick processes using spectral measures.
• Complexity analysis of the proposed and existing algorithms.

Main Results:

• The proposed extremal function algorithm provides exact simulations of max-stable processes.
• The new algorithm demonstrates superior computational efficiency compared to existing methods.
• Closed-form expressions are derived for implementing the algorithms for common models.

Conclusions:

• The extremal function approach offers a significant advancement in the exact simulation of max-stable processes.
• The developed methods enhance the practical application of extreme value theory in spatial statistics.
• An adaptive design is proposed for efficient simulation on dense grids.