the Cox-Ingersoll-Ross process
/ kox ING-er-sol ross /
The Cox-Ingersoll-Ross (CIR) process is a mean-reverting diffusion that, unlike the OU process, stays nonnegative — making it the standard model for quantities that must not go below zero, above all interest rates and stochastic variances. It answers the modelling need: I want mean reversion like the OU process, but I cannot tolerate negative values. The trick is to make the noise vanish as the process approaches zero, by putting a square root of the state into the diffusion coefficient.
It solves the SDE dX_t = a(b - X_t) dt + sigma sqrt(X_t) dB_t, with reversion speed a > 0, long-run mean b > 0, and volatility sigma. The drift a(b - X) is the same mean-reverting form as the OU process, but the diffusion sigma sqrt(X) shrinks to zero as X -> 0, so near the boundary the deterministic drift (which is positive when X is small) dominates and pushes the process back up. The square-root diffusion is not Lipschitz at 0, so the basic existence theorem does not apply directly; the Yamada-Watanabe square-root condition (|sqrt(x) - sqrt(y)| <= sqrt(|x-y|)) gives pathwise uniqueness, and a nonnegative solution exists. The boundary behaviour at 0 is governed by the Feller condition: if 2ab >= sigma^2 the process never reaches 0 (it stays strictly positive a.s.), whereas if 2ab < sigma^2 it touches 0 (and reflects, in the standard formulation). Its transition density is a non-central chi-squared, and its stationary distribution is a Gamma distribution. The CIR process is, up to scaling, a squared Bessel process subjected to mean reversion.
CIR is a cornerstone of mathematical finance: the CIR short-rate model and, as the Heston model's variance process, the workhorse of stochastic volatility. Its analytic tractability (affine term structure, closed-form bond and option transforms) plus its nonnegativity make it hugely useful. Honest cautions: the Feller condition 2ab >= sigma^2 cannot be dropped if you need a strictly positive process — when it fails, the process hits zero, which matters both for modelling (a variance that can hit exactly zero) and for numerics (naive Euler-Maruyama can produce negative values under the square root and must be modified, e.g. full-truncation or the Andersen QE scheme). And the square-root coefficient is exactly the borderline non-Lipschitz case where the standard SDE theory must be replaced by the Yamada-Watanabe refinement.
In the Heston stochastic-volatility model the instantaneous variance v_t follows a CIR process dv_t = kappa(theta - v_t) dt + xi sqrt(v_t) dW_t. If the Feller condition 2 kappa theta >= xi^2 holds, the variance stays strictly positive; if it fails, v_t can hit zero, and simulation schemes must be chosen carefully to avoid negative variances.
CIR as the Heston variance process: nonnegative, mean-reverting, but able to touch zero when Feller fails.
Nonnegativity is automatic, but strict positivity needs the Feller condition 2ab >= sigma^2; below it the process hits zero. The square-root diffusion is non-Lipschitz, so existence/uniqueness rests on Yamada-Watanabe, and naive Euler-Maruyama can go negative and must be modified.