Extreme values and the level-crossing problem: an application to the Feller process

Area of Science:

• Stochastic processes
• Probability theory
• Mathematical physics

Background:

• Understanding the behavior of random processes is crucial in various scientific fields.
• Extreme value analysis quantifies the limits of random phenomena.
• First-passage and escape problems are fundamental in analyzing random process dynamics.

Purpose of the Study:

• To review and analyze the extreme values attained by random processes.
• To connect the study of extreme values to level crossing problems (first-passage and escape).
• To provide detailed results for specific diffusion processes.

Main Methods:

• Relating extreme value analysis to first-passage and escape problems.
• Investigating maximum, minimum, maximum absolute value, and range of random processes.
• Specializing in diffusion processes, specifically the Wiener and Feller processes.

Main Results:

• Established connections between extreme values and boundary crossing phenomena.
• Detailed analysis of maximum, minimum, maximum absolute value, and range for diffusion processes.
• Specific results derived for the Wiener and Feller processes.

Conclusions:

• The study provides a comprehensive review of extreme values in random processes.
• Boundary crossing problems offer a valuable framework for understanding process extremes.
• The findings offer insights into the behavior of Wiener and Feller diffusion processes.