From CKLS Process to CIR-type Process: Using a Twice-differentiable Mapping and Generalized Girsanov's Theorem

Abstract

We construct a twice-differentiable mapping $T:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ satisfying $T^{\prime }(x)x^k=L\sqrt{T(x)}$ for a given constant $L$ and apply it to the CKLS short-rate process ${\lambda }_t$, which solves $d{\lambda }_t=(a-b{\lambda }_t)dt+\sigma {\lambda }_t^{k}d{W}_t$. By Itô's lemma, the transformed process $X_t=T({\lambda }_t)$ obeys an SDE whose diffusion is proportional to $\sqrt{X_t}$ and whose drift is nonlinear in ${\lambda }_t$. A critical review of an earlier study of this transformation reveals substantial errors in its model specification, derivations, and proofs. A generalized Girsanov change of measure then shifts the drift, yielding classical Cox-Ingersoll-Ross (CIR) dynamics for $X_t$ under an equivalent measure $\mathbb{Q}$. Using uniqueness, strong existence, and positivity of ${\lambda }_t$ established via the Yamada-Watanabe-Engelbert theorem and boundary analysis, we show that the combined mapping and Girsanov step is valid for $L,a,b,\sigma >0$ and $1/2<k<1$, the range of particular financial relevance, or $k=1/2$ with $2a\ge {\sigma }^2$, corresponding to CIR with Feller's condition satisfied. The CIR representation gives the transition density, moment formulas, stationary density, and boundary behavior of $X_t$, and the transition density of ${\lambda }_t$ under $\mathbb{Q}$. We explain why the available moment estimates alone do not establish Novikov's or Kazamaki's condition. Instead, we prove directly that the Doléans-Dade exponential associated with our Girsanov transformation is a true martingale on every finite time interval and hence defines a Radon-Nikodým derivative, validating the entire construction. Our argument adapts a result extending the classical martingale criterion: Feller's explosion test and boundary classification provide a necessary and sufficient condition for the stochastic exponential to be a true martingale.

Corrected CIR drift constant from ${\sigma }^2{L}^2/4$ to ${\sigma }^2{L}^2/2$ so Feller's condition holds; revised Girsanov kernel, boundary and martingale proofs, densities, and moment formulas. Completed CKLS positive and negative supremum moment bounds. Removed OU transformation and dependent claims. Expanded critique of Hu et al. (2015). CKLS-to-CIR framework retained. Title and abstract updated

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• A Generalized Mean-Reverting Equation and Applications