**3. Main Results**

In this section, we give the closed-form formula of conditional moments of processes (2) and (3). Applying the Feynman–Kac technique and assuming a special form of the conditional moment, we can express the solution of the resulting PDE as an infinite series and solve the system of recursive ODEs to obtain coefficients for the closed-form formula. The results for some special cases are also displayed.

In this work, under the probability measure P and *<sup>σ</sup>*−field F*<sup>t</sup>*, we first propose the integral-form formula for the conditional moment of an IND-CEV process for *γ* > 0:

$$\mu\_{\ell}^{\langle \gamma \rangle}(r, \tau) := \mathbb{E}\left[r\_T^{\gamma} \mid r\_t = r\right],\tag{4}$$

for all *r* > 0 and *τ* := *T* − *t* ∈ (0, *<sup>T</sup>*]. Obviously, *u<sup>γ</sup>* (*r*, 0) = *<sup>r</sup>γ*. The key idea involves a system with a recurrence differential equation that brings about the PDE by involving an asymmetric matrix. The form of PDE's solution associated with the conditional moment (4) is a polynomial expression motivated by [16,17,19–24]. Hence, we can solve its coefficients to obtain a closed-form formula directly.

**Theorem 2.** *Let rt be an IND-CEV process satisfying* (2)*. Assume that the γth conditional moment can be expressed in the form:*

$$\mu\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \sum\_{k=0}^{\infty} A\_{\ell}^{\langle k \rangle}(\tau) r^{\gamma - \ell k} \tag{5}$$

*in which the infinite series uniformly converges on D<sup>γ</sup>* ⊆ (0, ∞) × (0, *<sup>T</sup>*]*. Then, the coefficients in* (5) *can be expressed recursively by:*

$$\begin{aligned} A\_{\ell}^{\langle 0 \rangle}(\tau) &:= \mathbf{e}^{-\int\_0^{\tau} P\_{\ell}^{\langle 0 \rangle}(T - \xi) \mathrm{d}\xi}, \\ A\_{\ell}^{\langle k \rangle}(\tau) &:= \int\_0^{\tau} \mathbf{e}^{-\int\_{\eta}^{\tau} P\_{\ell}^{\langle k \rangle}(T - \xi) \mathrm{d}\xi} Q\_{\ell}^{\langle k - 1 \rangle}(T - \eta) A\_{\ell}^{\langle k - 1 \rangle}(\eta) \mathrm{d}\eta, \end{aligned} \tag{6}$$

*for all k* ∈ N*, where:*

$$P\_{\ell}^{\langle j \rangle}(\boldsymbol{\pi}) := (\gamma - \ell j)\mathbf{x}(\boldsymbol{\pi}),\tag{7}$$

$$Q\_{\ell}^{\langle j \rangle}(\tau) := (\gamma - \ell j) \left( \frac{1}{2} (\gamma - \ell j - 1) \sigma^2(\tau) + \kappa(\tau) \theta(\tau) \right). \tag{8}$$

**Proof.** Applying the Feynman–Kac formula to the SDE (2), we have that the function *u* := *u<sup>γ</sup>* (*r*, *τ*) satisfies the PDE:

$$u\_{\tau} - \frac{1}{2}\sigma^2 (T - \tau) r^{-(\ell - 2)} u\_{rr} - \kappa (T - \tau) \left(\theta (T - \tau) r^{-(\ell - 1)} - r\right) u\_{\tau} = 0 \tag{9}$$

for all *r* > 0 and 0 < *τ* ≤ *T*, with the initial condition:

$$\mu\_{\ell}^{\langle \gamma \rangle}(r, 0) = \mathbf{E}\left[r\_T^{\gamma} \mid r\_T = r\right] = r^{\gamma}. \tag{10}$$

From (5), *u<sup>γ</sup>* (*r*, 0) = ∞∑*k*=0 *A<sup>k</sup>* (0)*rγ*−*k*. Comparing this with (10) implies that *A*<sup>0</sup> (0) = 1 and *A<sup>k</sup>* (0) = 0 for all *k* ∈ N. Substituting (5) into (9), we have that:

$$\begin{split} 0 &= \sum\_{k=0}^{\infty} \frac{\mathbf{d}}{\mathbf{d}\tau} A\_{\ell}^{\langle k\rangle}(\tau) r^{\gamma-\ell k} \\ &- \frac{1}{2} \sigma^{2} (T-\tau) r^{-(\ell-2)} \sum\_{k=0}^{\infty} \left( (\gamma-\ell k)(\gamma-\ell k-1) A\_{\ell}^{\langle k\rangle}(\tau) r^{\gamma-\ell k-2} \right) \\ &- \kappa (T-\tau) \left( \theta(T-\tau) r^{-(\ell-1)} - r \right) \sum\_{k=0}^{\infty} \left( (\gamma-\ell k) A\_{\ell}^{\langle k\rangle}(\tau) r^{\gamma-\ell k-1} \right) \end{split}$$

or it can be simplified as:

$$\begin{split} 0 &= \left( \frac{\mathbf{d}}{\mathbf{d}\tau} A\_{\ell}^{\langle 0 \rangle} (\tau) + \gamma \kappa (T - \tau) A\_{\ell}^{\langle 0 \rangle} (\tau) \right) r^{\gamma} \\ &+ \sum\_{k=1}^{\infty} \left( \frac{\mathbf{d}}{\mathbf{d}\tau} A\_{\ell}^{\langle k \rangle} (\tau) + P\_{\ell}^{\langle k \rangle} (T - \tau) A\_{\ell}^{\langle k \rangle} (\tau) - Q\_{\ell}^{\langle k - 1 \rangle} (T - \tau) A\_{\ell}^{\langle k - 1 \rangle} (\tau) \right) r^{\gamma - \ell k} . \end{split}$$

Under the assumption that the solution is in the form (5) over *D<sup>γ</sup>* , this equation can be solved through the system of ODEs:

$$\begin{split} 0 &= \frac{\mathbf{d}}{\mathbf{d}\tau} A\_{\ell}^{\langle 0 \rangle}(\tau) + \gamma \kappa (T - \tau) A\_{\ell}^{\langle 0 \rangle}(\tau), \\ 0 &= \frac{\mathbf{d}}{\mathbf{d}\tau} A\_{\ell}^{\langle k \rangle}(\tau) + P\_{\ell}^{\langle k \rangle}(T - \tau) A\_{\ell}^{\langle k \rangle}(\tau) - Q\_{\ell}^{\langle k - 1 \rangle}(T - \tau) A\_{\ell}^{\langle k - 1 \rangle}(\tau), \end{split} \tag{11}$$

with initial conditions *A*<sup>0</sup> (0) = 1 and *A<sup>k</sup>* (0) = 0 for *k* ∈ N. Hence, the coefficients in the infinite series (5) can be directly acquired by solving the system (11), which turns out to be the recursive relation given in (6).

Note that when we define variables or notations using the := sign, e.g., Equations (6)–(8), we will use those variables or notations throughout this work.

Observe that (5) becomes a finite sum when one of the two factors for *Q<sup>j</sup>* (*τ*) in (8) is zero. For fixing > 0, we give the consequence of (5) in Theorem 2 when *γ*/ ∈ Z+. The infinite sum in (5) is cut off at a finite order and can be presented as in the following corollary.

**Corollary 1.** *Let rt be an IND-CEV process satisfying* (2)*. For the positive real number γ such that γ*/ ∈ Z+*, the γth conditional moment is explicitly given by:*

$$u\_{\ell}^{\langle \gamma \rangle}(r, \mathbf{r}) = \sum\_{k=0}^{\gamma/\ell} A\_{\ell}^{\langle k \rangle}(\mathbf{r}) r^{\gamma - \ell \mathbf{k}},\tag{12}$$

*for all* (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*]*.*

**Proof.** From (8), when *j* = *γ*/, we acquire that *Q<sup>j</sup>* (*τ*) = 0. From (6), the coefficients *A<sup>k</sup>* (*τ*) = 0 for all integers *k* ≥ *γ*/ + 1. Hence, the infinite sum (5) is actually just the finite sum (12). Since any integration of a continuous function over a compact set is finite, the finite sum (12) exists for all (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*]; hence, the infinite sum (5) uniformly converges to the finite sum (12) and *D<sup>γ</sup>* = (0, ∞) × (0, *<sup>T</sup>*].

Another consequence of (5) in Theorem 2 is shown in the following corollary.

**Corollary 2.** *Assume that rt follows SDE* (2) *and there exists m* ∈ <sup>Z</sup>+0*such that:*

$$\gamma = 1 - \frac{2\kappa(\tau)\theta(\tau)}{\sigma^2(\tau)} + \ell m \tag{13}$$

*for all τ* ∈ (0, *<sup>T</sup>*]*. Then,*

$$u\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \sum\_{k=0}^{m} A\_{\ell}^{\langle k \rangle}(\tau) r^{\gamma - \ell k},\tag{14}$$

*for all* (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*]*.*

**Proof.** From (8), when *j* = *m*, we have that *Q<sup>j</sup>* (*τ*) = 0. From (6), the coefficients *A<sup>k</sup>* (*τ*) = 0 for all integers *k* ≥ *m* + 1. With the same reasoning as in the proof of Corollary 1, we acquire the desired result.

One main concern when we investigate the conditional moments described by the IND-CEV process is that the integral terms (6) in Theorem 2 cannot be directly evaluated. Thus, a very accurate numerical integration scheme is applied via the Chebyshev integration method; see [25–28] for more details.

Next, we consider the case when *<sup>κ</sup>*(*τ*), *<sup>θ</sup>*(*τ*) and *σ*(*τ*) are constant functions.

**Theorem 3.** *If rt follows the SDE* (3) *and the γth conditional moment can be expressed in the form* (5)*, then the γth conditional moment is given by:*

$$
\mu\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \sum\_{k=0}^{\infty} \frac{\mathbf{e}^{-\gamma \kappa \tau}}{k!} \left( \frac{\mathbf{e}^{\kappa \tau \ell} - 1}{\kappa \ell} \right)^{k} \left( \prod\_{j=0}^{k-1} \breve{Q}\_{\ell}^{\langle j \rangle} \right) r^{\gamma - \ell k} \,, \tag{15}
$$

*for all* (*r*, *τ*) ∈ *D<sup>γ</sup> , where:*

$$
\tilde{Q}\_{\ell}^{\langle j \rangle} := \left( \gamma - \ell j \right) \left( \frac{1}{2} (\gamma - \ell j - 1) \sigma^2 + \kappa \theta \right). \tag{16}
$$

*Note that the product from* 0 *to* <sup>−</sup>1*,* −1 ∏ *j*=0 *Q* 5*j , is defined to be* 1*.*

**Proof.** We will prove by induction that:

$$A\_{\ell}^{\langle k \rangle}(\tau) = \frac{\mathbf{e}^{-\gamma \kappa \tau}}{k!} \left( \frac{\mathbf{e}^{\kappa \tau \ell} - 1}{\kappa \ell} \right)^{k} \left( \prod\_{j=0}^{k-1} \widetilde{Q}\_{\ell}^{\langle j \rangle} \right)^{\varepsilon}$$

for all *k* ∈ N ∪ {0}. From (6) with the constant parameters *κ*, *θ* and *σ*, we have that *A*<sup>0</sup> (*τ*) = e<sup>−</sup>*γκτ* and

$$A\_{\ell}^{\langle k \rangle}(\mathbf{r}) = \tilde{Q}\_{\ell}^{\langle k-1 \rangle} \int\_0^{\mathbf{r}} \mathbf{e}^{-(\mathbf{r}-\eta)(\gamma-\ell\mathbf{k})\mathbf{x}} A\_{\ell}^{\langle k-1 \rangle}(\eta) \mathbf{d}\eta,\tag{17}$$

for all *k* ∈ N. By substituting *k* = 1 in (17), we obtain:

$$A\_{\ell}^{\langle 1 \rangle}(\tau) = \mathbf{e}^{-\gamma \kappa \tau} \left( \frac{\mathbf{e}^{\kappa \tau \ell} - 1}{\kappa \ell} \right) \widetilde{Q}\_{\ell}^{\langle 0 \rangle}.$$

Let *k* ∈ N. Assume that:

$$A\_{\ell}^{\langle k-1 \rangle}(\tau) = \frac{\mathbf{e}^{-\gamma \kappa \tau}}{(k-1)!} \left( \frac{\mathbf{e}^{\kappa \tau \ell} - 1}{\kappa \ell} \right)^{k-1} \left( \prod\_{j=0}^{k-2} \tilde{Q}\_{\ell}^{\langle j \rangle} \right).$$

From (17), we have that:

$$\begin{split} A\_{\ell}^{\langle k\rangle}(\mathbf{r}) &= \mathbf{e}^{-(\gamma-\ell k)\mathbf{x}\tau} \widetilde{\mathbf{Q}}\_{\ell}^{\langle k-1\rangle} \int\_{0}^{\mathsf{T}} \mathbf{e}^{(\gamma-\ell k)\mathbf{x}\eta} A\_{\ell}^{\langle k-1\rangle}(\eta) \mathbf{d}\eta \\ &= \frac{\mathbf{e}^{-(\gamma-\ell k)\mathbf{x}\tau}}{(k-1)!(\kappa\ell)^{k-1}} \left( \prod\_{j=0}^{k-1} \widetilde{\mathbf{Q}}\_{\ell}^{\langle j\rangle} \right) \int\_{0}^{\mathsf{T}} \mathbf{e}^{-k\ell x\eta} \left( \mathbf{e}^{\mathbf{x}\eta\ell} - 1 \right)^{k-1} \mathbf{d}\eta \\ &= \frac{\mathbf{e}^{-\gamma\mathbf{x}\tau}}{k!} \left( \frac{\mathbf{e}^{\mathbf{x}\tau\ell} - 1}{\kappa\ell} \right)^{k} \left( \prod\_{j=0}^{k-1} \widetilde{\mathbf{Q}}\_{\ell}^{\langle j\rangle} \right). \quad \mathsf{D} \end{split}$$

From Corollaries 1 and 2, when *<sup>κ</sup>*(*τ*), *<sup>θ</sup>*(*τ*) and *σ*(*τ*) are constant functions, we have the following corollaries.

**Corollary 3.** *Assume that rt follows SDE* (3)*. For a positive real number γ such that γ*/ ∈ Z+*, the γth conditional moment is explicitly given by:*

$$\mu\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \sum\_{k=0}^{\gamma/\ell} \frac{\mathbf{e}^{-\gamma \chi \tau}}{k!} \left( \frac{\mathbf{e}^{\kappa \tau \ell} - 1}{\kappa \ell} \right)^{k} \left( \prod\_{j=0}^{k-1} \tilde{\mathbf{Q}}\_{\ell}^{\langle j \rangle} \right) r^{\gamma - \ell k} \,, \tag{18}$$

*for all* (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*]*. Note that the product of Q* 5*j in* (18) *for k* = 0 *is defined to be 1.*

**Corollary 4.** *Assume that rt follows the SDE* (3)*. If there exists m* ∈ <sup>Z</sup>+0*such that*

$$\gamma = 1 - \frac{2\kappa\theta}{\sigma^2} + \ell m\_\prime \tag{19}$$

*then*

$$\mu\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \sum\_{k=0}^{m} \frac{\mathbf{e}^{-\gamma \mathbf{x} \tau}}{k!} \left( \frac{\mathbf{e}^{\mathbf{x} \tau \ell} - 1}{\kappa \ell} \right)^{k} \left( \prod\_{j=0}^{k-1} \widetilde{Q}\_{\ell}^{\langle j \rangle} \right) r^{\gamma - \ell k},\tag{20}$$

*for all* (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*]*.*

> For SDE (3), characterization for the convergence of the series (15) can be provided.

**Theorem 4.** *Assume that rt follows SDE* (3) *and Q* 5*j* = 0 *for all j* ∈ <sup>Z</sup>+0 *. Then, the series* (15) *diverges for all* (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*]*.*

**Proof.** Since *Q* 5*j* = 0 for all *j* ∈ <sup>Z</sup>+0 , we have that *γ* − *k* = 0 and (*γ* − *k* −<sup>1</sup>)*σ*2/2+*κθ* = 0 for all *k* ∈ <sup>Z</sup>+0 .

$$\begin{split} \lim\_{k \to \infty} \left| \frac{A\_{\ell}^{(k+1)}(\tau)r^{\gamma-\ell(k+1)}}{A\_{\ell}^{(k)}(\tau)r^{\gamma-\ell k}} \right| &= \lim\_{k \to \infty} \left| \frac{\frac{\mathbf{e}^{-\gamma\pi\tau}}{(k+1)!} \left(\frac{\mathbf{e}^{\pi\ell\ell}-1}{\kappa\ell}\right)^{k+1} \left(\prod\_{j=0}^{k} \widetilde{Q}\_{\ell}^{(j)}\right) r^{\gamma-\ell(k+1)}}{\frac{\mathbf{e}^{-\gamma\pi\tau}}{k!} \left(\frac{\mathbf{e}^{\pi\ell\ell}-1}{\kappa\ell}\right)^{k} \left(\prod\_{j=0}^{k-1} \widetilde{Q}\_{\ell}^{(j)}\right) r^{\gamma-\ell k}} \right| \\ &= \lim\_{k \to \infty} \left| \frac{\left(\mathbf{e}^{\pi\tau\ell}-1\right)\left(\gamma-\ell k\right) \left(\frac{1}{2}(\gamma-\ell k-1)\sigma^{2} + \kappa\theta\right)}{(k+1)\kappa\ell r^{\ell}} \right|. \end{split}$$

The above expression is O(*k*) as *k* → ∞; hence, the limit diverges. By ratio test, the series (15) diverges for all (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*].

From Corollaries 3 and 4, and Theorem 4, we have the following result.

**Corollary 5.** *Assume that rt follows SDE* (3)*. Then, the series* (15) *converges for all* (*r*, *τ*) ∈ (0, ∞) × (0, *T*] *if and only if:*

*1. γ*∈ Z+*, or*

*2.* 1*γ* − 1 + 2*κθ σ*<sup>2</sup> ∈ <sup>Z</sup>+0*.*

*The convergent results for case* 1 *and* 2 *are given in Corollaries 3 and 4, respectively.*
