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First-passage and escape problems in the Feller process.

Jaume Masoliver1, Josep Perelló

  • 1Departament de Física Fonamental, Universitat de Barcelona, Diagonal, 647, E-08028 Barcelona, Spain. jaume.masoliver@ub.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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The Feller process, a diffusion model, is explored for its lesser-known level-crossing properties. This study analyzes first-passage and escape problems for this versatile mathematical tool.

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Area of Science:

  • Stochastic Processes
  • Mathematical Finance
  • Computational Neuroscience

Background:

  • The Feller process is a one-dimensional diffusion model characterized by linear drift and a diffusion coefficient dependent on the state, which becomes zero at the origin.
  • Its positive definite nature and linear characteristics make it suitable for modeling diverse phenomena, including single-neuron firing and financial asset volatility.
  • While general properties are established, specific behaviors like first-passage and escape problems related to level crossing remain less understood.

Purpose of the Study:

  • To thoroughly investigate and elucidate the properties of the Feller process concerning level crossing phenomena.
  • To address the less-explored first-passage and escape problems associated with the Feller process.
  • To provide a comprehensive analysis of these specific aspects of the Feller process.

Main Methods:

  • Analysis of stochastic differential equations governing the Feller process.
  • Mathematical techniques for studying first-passage time distributions.
  • Methods for analyzing escape probabilities in diffusion processes.

Main Results:

  • Detailed characterization of first-passage time distributions for the Feller process.
  • Quantification of escape probabilities from specific regions.
  • New insights into the boundary behavior of the Feller process.

Conclusions:

  • The study provides a comprehensive understanding of Feller process level-crossing properties, specifically first-passage and escape problems.
  • The findings enhance the applicability of the Feller process in fields requiring detailed analysis of hitting and exit times.
  • This work contributes to the theoretical framework of diffusion processes with vanishing diffusion coefficients.