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

Abstract We construct a twice-differentiable mapping $$T(x): \mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$$ satisfying $$\frac{dT(x)}{dx}x^{k}=L\big [T(x)\big ]^{\frac{1}{2}}$$ for a given constant L and apply it to the CKLS short-rate process $$\lambda _{t}$$ , which solves the stochastic differential equation (SDE) of the form $$d\lambda _{t}=(a-b\lambda _{t})dt+\sigma (\lambda _{t})^{k}dW_{t}$$ . By Itô’s lemma,the transformed process $$X_{t}\overset{\text {def}}{=}T(\lambda _{t})$$ obeys an SDE whose diffusion term is proportional to $$(\lambda _{t})^{\frac{1}{2}}$$ and whose drift is a non-linear function of $$\lambda _{t}$$ . A critical review of an earlier study on the same transformation reveals substantial errors in its model specification, derivations, and proofs. Next, a generalized Girsanov transformation of measure is introduced to shift the drift. Under the equivalent measure $$\mathbb {Q}$$ , the dynamics of $$X_{t}$$ reduces to the classical Cox–Ingersoll–Ross (CIR) form. Leveraging well-known properties concerning uniqueness, strongness, and positivity of $$\lambda _{t}$$ induced by the Yamada-Watanabe-Engelbert theorem, we show that the combined twice-differentiable mapping and Girsanov step is valid precisely when $$L>0$$ , $$a>0, b>0, \sigma>0$$ and, most importantly, $$\frac{1}{2}<k<1$$ (which is the parameter range of particular relevance in financial applications) or $$k=\frac{1}{2}$$ with $$2a\ge \sigma ^2$$ (which reduces to the CIR process with Feller’s condition satisfied). The CIR representation allows us to import a suite of results including stationary density, moment formulas, and boundary behavior, and, by further mapping to an Ornstein–Uhlenbeck framework ensured by the specific relationship between the coefficients of the two SDEs, to derive additional distributional properties of $$\lambda _{t}$$ under $$\mathbb {Q}$$ , including explicit expressions of the transition density, moment generating function and the SDE, respectively. Finally, we demonstrate why the classical Novikov’s and Kazamaki’s conditions cannot be verified, and then prove directly that the Doléans-Dade exponential associated with our Girsanov transformation is a true martingale (thus can be called Radon-Nikodym derivative), thus we have the soundness of the entire procedure combining T ( x ) and the Girsanov transformation validated. Our argument adapts a recent result, rather than relying on Novikov’s or Kazamaki’s conditions, that extends the classical martingale criterion: by applying Feller’s explosion test together with his boundary classification, it supplies a necessary and sufficient condition under which the Radon–Nikodym derivative is a true martingale.