**2. IND-CEV Process**

This section presents the IND-CEV process and sufficient assumptions for the process in order to have a unique positive solution. The dynamics of the short-term interest rate over time are assumed to follow the SDE:

$$\mathbf{d}r\_t = \kappa(t) \left( \theta(t) r\_t^{2\beta - 1} - r\_t \right) \mathbf{d}t + \sigma(t) r\_t^{\beta} \mathbf{d}\mathcal{W}\_{t\prime} \tag{1}$$

with the initial condition *r*0 > 0, where *<sup>κ</sup>*(*t*), *θ*(*t*) and *σ*(*t*) are smooth and bounded time-dependent parameter functions and *Wt* is a standard Brownian motion, which has asymmetric sample paths, under a probability space (<sup>Ω</sup>, F,P) with filtration {F*t*}*t*≥0. In this study, we only focus on the case that *β* < 1 in the SDE (1). Let := 2 − 2*β*. Henceforth, the dynamics of the process *rt* are considered via the following SDE:

$$\mathbf{d}r\_t = \kappa(t) \left( \theta(t) r\_t^{-(\ell-1)} - r\_t \right) \mathbf{d}t + \sigma(t) r\_t^{-\left(\frac{\ell-2}{2}\right)} \mathbf{d}W\_t \tag{2}$$

where > 0. The process *rt* in (2) is called an IND-CEV process. In addition, the SDE (2) is called the extended Cox–Ingersoll–Ross (ECIR) process when = 1; see for more details in [14–17]. From (2), if the parameters *<sup>κ</sup>*(*t*), *θ*(*t*) and *σ*(*t*) are constants written by *κ*, *θ* and *σ*, respectively, then the SDE (2) can be rewritten as:

$$\mathbf{d}r\_t = \kappa \left(\theta r\_t^{-\left(\ell-1\right)} - r\_t\right)\mathbf{d}t + \sigma r\_t^{-\left(\frac{\ell-2}{2}\right)}\mathbf{d}\mathcal{W}\_t \tag{3}$$

where > 0. We will consider SDEs (2) and (3) on a time domain [0, *<sup>T</sup>*].

We first discuss the solution of SDE (2). **Assumption 1.** *The parameter functions <sup>θ</sup>*(*t*), *κ*(*t*) *and σ*(*t*) *in SDE* (2) *are strictly positive and continuously differentiable on* [0, *<sup>T</sup>*]*. Moreover, κ*(*t*)/*σ*<sup>2</sup>(*t*) *is locally bounded on* [0, *<sup>T</sup>*]*.*

**Assumption 2.** <sup>2</sup>*κ*(*t*)*θ*(*t*) > *σ*<sup>2</sup>(*t*) *for all t* ∈ [0, *<sup>T</sup>*]*.*

**Theorem 1.** *For SDE* (2)*, if Assumptions 1 and 2 hold with r*0 > 0*, then there exists a pathwise unique strong solution process rt* > 0 *for all t* ∈ [0, *<sup>T</sup>*]*.*

**Proof.** Transforming *vt* = *rt* with the Itô lemma yields:

$$\begin{split} \mathrm{d}\mathbf{v}\_{l} &= (\ell)r\_{l}^{\ell-1} \Big( \kappa(t) \Big( \theta(t)r\_{l}^{-(\ell-1)} - r\_{l} \Big) \mathrm{d}t + \sigma(t)r\_{l}^{-\frac{\ell-2}{2}} \mathrm{d}\mathcal{W}\_{l} \Big) + \frac{1}{2}(\ell)(\ell-1)r\_{l}^{\ell-2} \Big( \sigma(t)r\_{l}^{-\frac{\ell-2}{2}} \mathrm{d}\mathcal{W}\_{l} \Big)^{2} \\ &= \Big( \ell \mathbf{x}(t) \Big( \theta(t) - r\_{l}^{\ell} \Big) + \frac{1}{2}(\ell)(\ell-1)\sigma^{2}(t) \Big) \mathrm{d}t + \ell \sigma(t)r\_{l}^{\frac{\ell}{2}} \mathrm{d}\mathcal{W}\_{l} \\ &= \ell \mathbf{x}(t) \Big( \theta(t) - r\_{l}^{\ell} + \frac{(\ell-1)\sigma^{2}(t)}{2\kappa(t)} \Big) \mathrm{d}t + \ell \sigma(t)r\_{l}^{\frac{1}{2}\ell} \mathrm{d}\mathcal{W}\_{l} \\ &= \ell \mathbf{x}(t) \Big( \theta(t) + \frac{(\ell-1)\sigma^{2}(t)}{2\kappa(t)} - v\_{l} \Big) \mathrm{d}t + \ell \sigma(t)\sqrt{v\_{l}} \mathrm{d}\mathcal{W}\_{l} \\ &= A\_{\ell}(t)(B\_{\ell}(t) - v\_{l}) \mathrm{d}t + \mathcal{C}\_{\ell}(t)\sqrt{v\_{l}} \mathrm{d}\mathcal{W}\_{l}, \end{split}$$

where *<sup>A</sup>*(*t*) = *κ*(*t*), *<sup>B</sup>*(*t*) = *θ*(*t*)+( − <sup>1</sup>)*σ*<sup>2</sup>(*t*)/2*κ*(*t*) and *<sup>C</sup>*(*t*) = *σ*(*t*). Thus, *vt* is an ECIR process. Under Assumptions 1 and 2, the functions *A*, *B* and *C* are strictly positive, smooth and continuous time-dependent parameter functions on [0, *<sup>T</sup>*]. Additionally, we have that:

$$\begin{aligned} 2A\_\ell(t)B\_\ell(t) &= 2\ell \kappa(t) \left( \theta(t) + \frac{(\ell-1)\sigma^2(t)}{2\kappa(t)} \right) \\ &= \ell \left( 2\kappa(t)\theta(t) + (\ell-1)\sigma^2(t) \right) \\ &> \ell \left( \sigma^2(t) + (\ell-1)\sigma^2(t) \right) = \mathcal{C}\_\ell^2(t). \end{aligned}$$

By the Feller condition [18], the SDE (2) has a pathwise unique strong solution in which *vt* avoids zero almost surely under measure P for all 0 < *t* ≤ *T* and so does *rt*.

From now on, we will always assume Assumptions 1 and 2 with *r*0 > 0.
