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Driven by Brownian motion Cox-Ingersoll-Ross and squared Bessel processes: interaction and phase transition

Yuliya Mishura, Kostiantyn Ralchenko, Svitlana Kushnirenko

Abstract

This paper studies two related stochastic processes driven by Brownian motion: the Cox-Ingersoll-Ross (CIR) process and the Bessel process. We investigate their shared and distinct properties, focusing on time-asymptotic growth rates, distance between the processes in integral norms, and parameter estimation. The squared Bessel process is shown to be a phase transition of the CIR process and can be approximated by a sequence of CIR processes. Differences in stochastic stability are also highlighted, with the Bessel process displaying instability, while the CIR process remains ergodic and stable.

Driven by Brownian motion Cox-Ingersoll-Ross and squared Bessel processes: interaction and phase transition

Abstract

This paper studies two related stochastic processes driven by Brownian motion: the Cox-Ingersoll-Ross (CIR) process and the Bessel process. We investigate their shared and distinct properties, focusing on time-asymptotic growth rates, distance between the processes in integral norms, and parameter estimation. The squared Bessel process is shown to be a phase transition of the CIR process and can be approximated by a sequence of CIR processes. Differences in stochastic stability are also highlighted, with the Bessel process displaying instability, while the CIR process remains ergodic and stable.

Paper Structure

This paper contains 15 sections, 16 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Some historical information
  3. Overview of the results presented in the paper
  4. Structure of the paper
  5. Preliminaries
  6. Distributional and path-wise properties of CIR process
  7. Distributional and path-wise properties of squared Bessel process
  8. Time-asymptotic growth rate for CIR and quadratic Bessel processes
  9. Distance between CIR and squared Bessel processes in integral norms
  10. Parameter estimation
  11. Instability and some functional limit theorems for the squared Bessel process
  12. Appendix
  13. Special functions
  14. A limit theorem for equations with nonhomogeneous coefficients and non-Lipschitz diffusion
  15. Functional and other limit theorems for the solutions of stochastic differential equations

Key Result

Proposition 3.1

Let $Z = \{Z_t, t \ge 0\}$ be a unique solution to CIR or BES (i.e., $Z = X$ or $Z = Y$). Then, for all $t \ge 0$,

Theorems & Definitions (41)

  • Remark 2.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Theorem 4.1
  • Remark 4.2
  • ...and 31 more