The CIR Transition Distribution and Its Fokker–Planck Derivation

Summary

The document asks for a complete derivation of the transition distribution of the Cox–Ingersoll–Ross process. It notes the commonly stated result that the distribution is noncentral chi-squared and identifies the Kolmogorov forward, or Fokker–Planck, equation as a possible route to the solution. The discussion also points to Feller’s work on the relevant partial differential equation and highlights the case in which the Feller condition holds.

No derivation or reference that fills the gap is provided. Instead, the author describes difficulty finding an account that makes the transition from the forward equation to the distribution clear, and says that other treatments appear incomplete or use different methods. The document therefore serves mainly as a statement of a mathematical research question. It signals the distributional result and the derivation challenge, but gives no proof, assumptions beyond the stated condition, or numerical or empirical evidence.

Key ideas

• The CIR process transition distribution is commonly described as a noncentral chi-squared distribution.
• The Kolmogorov forward equation is identified as one possible path to deriving that distribution.
• The question highlights the derivation under the Feller condition as a point needing explanation.
• The document does not supply a derivation, proof, or complete supporting reference.
• Its value is identifying a technical gap rather than presenting a finished method.

Tags

# Derivation of the distribution for a CIR process


# Derivation of the distribution for a CIR process












Where is it possible to find a complete derivation of the distribution of a CIR process?

There is a number of papers that claim that it is a noncentrical chi-squared distribution. However, I cannot find anything where a detailed explanation of its derivation is given. The starting point is the paper by Feller (1951) that deals with the PDE that is exactly the Kolmogorov Forward Equation / Fokker-Planck Equation for the CIR model. Nevertheless, that paper does not provide any explicit solution in the case where the so-called "Feller's condition" holds. Other papers look incomplete or use other approaches. While the the most famous one (i.e. the transition probability derivation from a Kolgomogorov equation) seems unclear.

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