**4. Probabilistic Properties**

This section illustrates some usefulness of our results from Section 3 including the first, second and fractional conditional moments; conditional variance and central moments; conditional mixed moments; and conditional covariance and correlation.

**Example 1** (The conditional moments)**.** *From Corollary 1, the nth conditional moment of an IND-CEV process when the parameter* = 1/*L for some L* ∈ N *is given by:*

$$\mathbb{E}[r\_T^n \mid r\_t = r] = \mu\_\ell^{\langle n \rangle}(r, \tau) = \sum\_{k=0}^{nL} A\_\ell^{\langle k \rangle}(\tau) r^{n-k},$$

*where:*

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

*for k* ∈ N*, where:*

$$\begin{aligned} P\_\ell^{\langle j \rangle}(\tau) &= \left( n - \frac{j}{L} \right) \kappa(\tau), \\ Q\_\ell^{\langle j \rangle}(\tau) &= \left( n - \frac{j}{L} \right) \left( \frac{1}{2} \left( n - \frac{j}{L} - 1 \right) \sigma^2(\tau) + \kappa(\tau) \theta(\tau) \right). \end{aligned}$$

*For constants κ, θ and σ, we use u*<sup>1</sup> (*r*, *τ*) *and u*<sup>2</sup> (*r*, *τ*) *in Corollary 3. Then, for L* = 1*, the first and second conditional moments are given by:*

$$\mathbb{E}[r\_T \mid r\_t = r] = (r - \theta)\mathbf{e}^{-\kappa \tau} + \theta \tag{21}$$

*and*

$$\mathbf{E}\left[r\_T^2 \mid r\_t = r\right] = \mathbf{e}^{-2\mathbf{x}\tau}r^2 + \frac{\left(\sigma^2/2 + \kappa\theta\right)\mathbf{e}^{-2\mathbf{x}\tau}}{\kappa}\left(r(\mathbf{e}^{\mathbf{x}\tau} - 1) + \theta(\mathbf{e}^{\mathbf{x}\tau} - 1)^2\right). \tag{22}$$

*For L* = 2*, the first and second conditional moments are given by:*

$$\mathbf{E}[r\_T \mid r\_t = r] = \mathbf{e}^{-\kappa \tau} \left( r + \theta \left( \mathbf{e}^{\frac{\kappa \tau}{2}} - 1 \right) \left( 2r^{\frac{1}{2}} + \frac{\left( \mathbf{e}^{\frac{\kappa \tau}{2}} - 1 \right)}{\kappa} \left( -\frac{\sigma^2}{4} + \kappa \theta \right) \right) \right) \tag{23}$$

*and*

$$\begin{split} \mathbf{E}\left[r\_{T}^{2}\mid r\_{I}=r\right] &= \mathbf{e}^{-2\kappa\tau} \left(r^{2} + \left(\mathbf{e}^{\frac{\kappa\tau}{2}} - 1\right) \left(\frac{\sigma^{2}}{2} + \kappa\theta\right) \left(\frac{4}{\kappa}r^{\frac{3}{2}} + \frac{6\left(\mathbf{e}^{\frac{2\tau}{2}} - 1\right)}{\kappa^{2}} \left(\frac{\sigma^{2}}{4} + \kappa\theta\right)r\right)\right) \\ &+ \mathbf{e}^{-2\kappa\tau} \frac{4\left(\mathbf{e}^{\frac{\kappa\tau}{2}} - 1\right)^{3}}{\kappa^{2}} \left(\frac{\sigma^{2}}{2} + \kappa\theta\right) \left(\frac{\sigma^{2}}{4} + \kappa\theta\right)\theta r^{\frac{1}{4}} \\ &+ \mathbf{e}^{-2\kappa\tau} \frac{\left(\mathbf{e}^{\frac{\kappa\tau}{2}} - 1\right)^{4}}{\kappa^{2}} \left(\frac{\sigma^{2}}{2} + \kappa\theta\right) \left(\frac{\sigma^{2}}{4} + \kappa\theta\right) \left(-\frac{\sigma^{2}}{4} + \kappa\theta\right)\theta. \end{split} \tag{24}$$

*Additionally, for* = 3/4*, the conditional moment with γ* = 3/2 *is given by:*

$$\mathrm{E}\left[r\_{T}^{\frac{3}{2}}\mid r\_{t}=r\right] = \mathrm{e}^{-\frac{3}{2}\kappa\tau}r^{\frac{3}{2}} + 2\mathrm{e}^{-\frac{3}{2}\kappa\tau}\left(\frac{\mathrm{e}^{\frac{3}{2}\kappa\tau}-1}{\kappa}\right)\left(\frac{\sigma^{2}}{4}+\kappa\theta\right)r^{\frac{3}{4}}$$

$$+\mathrm{e}^{-\frac{3}{2}\kappa\tau}\left(\frac{\mathrm{e}^{\frac{3}{2}\kappa\tau}-1}{\kappa}\right)^{2}\left(\frac{\sigma^{2}}{4}+\kappa\theta\right)\left(-\frac{\sigma^{2}}{8}+\kappa\theta\right). \tag{25}$$

Next, we propose the consequences of Example 1, which are the conditional variance and central moments, conditional mixed moments, and conditional covariance and correlation, as follows.

**Example 2** (The conditional variance and *nth* central moment)**.** *By applying Corollary 3,* (21) *and* (22)*, the conditional variance of the IND-CEV process can be given by:*

$$\mathbf{Var}[r\_T|r\_t=r] = \mathbf{E}\left[\left(r\_T - \mathbf{E}[r\_T \mid r\_t]\right)^2 \mid r\_t = r\right] = u\_\ell^{(2)}(r,\tau) - \left(u\_\ell^{(1)}(r,\tau)\right)^2,$$

*where u*<sup>1</sup> (*r*, *τ*) *and u*<sup>2</sup> (*r*, *τ*) *are derived in* (21) *and* (22) *for the CIR process. In general, the nth central moment is presented by:*

$$\mu\_n(r,\tau) := \mathbf{E}\left[ (r\_T - \mathbf{E}[r\_T \mid r\_t])^n \mid r\_t = r \right] = \sum\_{j=0}^n (-1)^{n-j} \binom{n}{j} \left( u\_\ell^{(j)}(r,\tau) \right) \left( u\_\ell^{(1)}(r,\tau) \right)^{n-j}$$

*where u*<sup>0</sup> (*r*, *τ*) := 1*.*

**Example 3** (The conditional mixed moments)**.** *By applying the tower property for* 0 ≤ *t* < *T*1 < *T*2, *where τ*1 = *T*1 − *t and τ*2 = *T*2 − *T*1 *and Corollary 1, the conditional mixed moment of the IND-CEV process* (2) *with* = 1/*L is given by:*

$$\begin{split} \mathbf{E}\left[r\_{T\_{1}}^{n\_{1}}r\_{T\_{2}}^{n\_{2}}\mid r\_{t}=r\right] &= \mathbf{E}\left[r\_{T\_{1}}^{n\_{1}}\mathbf{E}\left[r\_{T\_{2}}^{n\_{2}}\mid r\_{T\_{1}}\right]\mid r\_{t}=r\right] = \mathbf{E}\left[r\_{T\_{1}}^{n\_{1}}u\_{\ell}^{\langle n\_{2}\rangle}\left(r\_{T\_{1}}, T\_{2}-T\_{1}\right)\mid r\_{t}=r\right] \\ &= \sum\_{k=0}^{n\_{2}L}A\_{\ell}^{(k)}\left(r\_{2}\right)\mathbf{E}\left[r\_{T\_{1}}^{n\_{1}+n\_{2}-\frac{k}{L}}\mid r\_{t}=r\right] \\ &= \sum\_{k=0}^{n\_{2}L}A\_{\ell}^{(k)}\left(r\_{2}\right)u\_{\ell}^{\langle n\_{1}+n\_{2}-\frac{k}{L}\rangle}\left(r, T\_{1}-t\right) \\ &= \sum\_{k=0}^{n\_{2}L}\sum\_{j=0}^{(n\_{1}+n\_{2})L-k}A\_{\ell}^{(k)}\left(r\_{2}\right)A\_{\ell}^{(j)}\left(r\_{1}\right)r^{n\_{1}+n\_{2}-\frac{k+j}{L}}. \end{split} \tag{26}$$

*In addition, the general formula for conditional mixed moments* **<sup>E</sup>**'*rn*<sup>1</sup> *T*1 *rn*<sup>2</sup> *T*2 ···*rnk Tk* | *rt* = *r*(*, where n*1, *n*2, ... , *nk* ∈ Z<sup>+</sup> *and* 0 ≤ *t* < *T*1 < *T*2 < ··· < *Tk, for the process* (3) *can be analytically derived by using Corollary 3.*

**Example 4** (The conditional covariance and correlation)**.** *The conditional covariance of the CIR process for* 0 ≤ *t* < *T*1 < *T*2, *where τ*1 = *T*1 − *t and τ*2 = *T*2 − *T*1, *is given by:*

$$\begin{split} \mathbf{Cov}\left[r\_{T\_1}, r\_{T\_2} \mid r\_t = r\right] &:= \mathbf{E}\left[\left(r\_{T\_1} - \mathbf{E}\left[r\_{T\_1} \mid r\_t\right]\right)\left(r\_{T\_2} - \mathbf{E}\left[r\_{T\_2} \mid r\_t\right]\right) \mid r\_t = r\right] \\ &= \mathbf{E}\left[r\_{T\_1} r\_{T\_2} \mid r\_t = r\right] - \mathbf{E}\left[r\_{T\_1} \mid r\_t = r\right] \mathbf{E}\left[r\_{T\_2} \mid r\_t = r\right] \\ &= \sum\_{k=0}^{1} \sum\_{j=0}^{2-k} A\_{\ell}^{(k)}(\tau\_2) A\_{\ell}^{(j)}(\tau\_1) r^{2-k-j} - u\_{\ell}^{(1)}(r, \tau\_1) u\_{\ell}^{(2)}(r, \tau\_2) .\end{split} \tag{27}$$

*Applying the results from* (26) *and* (27)*, we obtain that the conditional correlation of the CIR process is given by:*

$$\begin{split} \mathbf{Corr}[r\_{\overline{\tau}\_{1}}r\_{\overline{\tau}\_{2}}\mid r\_{l}=r]: &= \frac{\mathbf{Cov}[r\_{\overline{\tau}\_{1}},r\_{\overline{\tau}\_{2}}\mid r\_{l}=r]}{\mathbf{Var}[r\_{\overline{\tau}\_{1}}\mid r\_{l}=r]^{1/2}\mathbf{Var}[r\_{\overline{\tau}\_{2}}\mid r\_{l}=r]^{1/2}} \\ &= \frac{\sum\_{k=0}^{1}\sum\_{j=0}^{2-k} A\_{\ell}^{(k)}(\tau\_{2})A\_{\ell}^{(j)}(\tau\_{1})r^{2-k-j} - u\_{\ell}^{(1)}(r,\tau\_{1})u\_{\ell}^{(2)}(r,\tau\_{2})}{\left(u\_{1}^{(2)}(r,\tau\_{1}) - \left(u\_{1}^{(1)}(r,\tau\_{1})\right)^{2}\right)^{1/2}\left(u\_{1}^{(2)}(r,\tau\_{2}) - \left(u\_{1}^{(1)}(r,\tau\_{2})\right)^{2}\right)^{1/2}} . \end{split} \tag{28}$$

*We can generalize* (27) *and* (28) *by using* (26) *as the closed forms of* **Cov**'*rn*<sup>1</sup> *T*1 ,*rn*<sup>2</sup> *T*2 | *rt* = *r*( *and* **Corr**'*rn*<sup>1</sup> *T*1 ,*rn*<sup>2</sup> *T*2 | *rt* = *r*(*, where n*1 *and n*2 *are positive integers.*

### **5. Unconditional Moments of the IND-CEV Process**

This section provides the formula of the unconditional moments of the IND-CEV process with constant parameters as *τ* → ∞ reduced from the formula of conditional moments.

**Theorem 5.** *Assume that rt follows SDE* (3)*. Then, for all γ*/ ∈ Z+,

$$\lim\_{\tau \to \infty} u\_\ell^{\langle \gamma \rangle} (r, \tau) = \prod\_{j=1}^{\gamma/\ell} \frac{2\kappa \theta + (\ell j - 1)r^2}{2\kappa}. \tag{29}$$

**Proof.** Let *s* = *γ*/ ∈ Z+. By considering (18) in Corollary 3, the coefficient terms of *rγ*−*<sup>k</sup>* converge to 0 as *τ* → ∞ for *k* = 0, 1, 2, ... ,*s* − 1. Thus, the summation (18) is reduced to only one term, where *k* = *s*,

$$\begin{split} \lim\_{\tau \to \infty} u\_{\ell}^{\langle \gamma \rangle}(r, \tau) &= \lim\_{\tau \to \infty} \frac{\mathbf{e}^{-\gamma \kappa \tau}}{\mathbf{s}!} \left( \frac{\mathbf{e}^{\kappa \tau \ell} - 1}{\kappa \ell} \right)^{\delta} \left( \prod\_{j=0}^{s-1} \widetilde{Q}\_{\ell}^{(j)} \right) r^{\gamma - \ell s} \\ &= \frac{1}{\mathbf{s}! (\kappa \ell)^{s}} \left( \prod\_{j=0}^{s-1} \widetilde{Q}\_{\ell}^{(j)} \right) \lim\_{\tau \to \infty} \mathbf{e}^{-\gamma \kappa \tau} \left( \mathbf{e}^{\kappa \tau \ell} - 1 \right)^{s} \\ &= \frac{1}{\mathbf{s}! (\kappa \ell)^{s}} \left( \prod\_{j=0}^{s-1} \widetilde{Q}\_{\ell}^{(j)} \right) \lim\_{\tau \to \infty} \left( 1 - \mathbf{e}^{-\kappa \tau \ell} \right)^{s} \\ &= \frac{1}{\mathbf{s}! (\kappa \ell)^{s}} \left( \prod\_{j=0}^{s-1} \widetilde{Q}\_{\ell}^{(j)} \right), \end{split}$$

where *Q* 5*j* is defined in (16). By expressing *Q* 5*j* to the above equation, it can be performed to

$$\lim\_{\tau \to \infty} u\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \frac{1}{s! (\kappa \ell)^s} \prod\_{j=0}^{s-1} \left( \gamma - \ell j \right) \left( \frac{1}{2} (\gamma - \ell j - 1) \sigma^2 + \kappa \theta \right) \\ = \prod\_{j=1}^{\gamma/\ell} \frac{2 \kappa \theta + (\ell j - 1) \sigma^2}{2 \kappa}. \quad \Box$$

Note that the formula for unconditional moments does not rely on the initial value *r*, and these unconditional moments represent the moments of the stationary distribution of the process (3).
