**6. Experimental Validation**

In this section, we validate the closed-form formulas presented in Theorem 2 and Corollaries 1 and 2. The Euler–Maruyama (EM) method was applied to simulate the process (2) and approximate the conditional moments based on the symmetry concept. For an interval [0, *<sup>τ</sup>*], let Δ = *τ*/*N* for a fixed *N* ∈ N and *ti* = Δ*i* for *i* = 0, 1, ... , *N*. We denote a numerical solution of the IND-CEV process at time *ti* by *r*ˆ*ti* . The EM approximation of (2) on the interval [0, *τ*] is defined as *r*ˆ0 = *r* and

$$\mathfrak{H}\_{t\_{i+1}} = \mathfrak{H}\_{t\_i} + \kappa(t\_i) \left( \theta(t\_i) \mathfrak{f}\_{t\_i}^{-(\ell-1)} - \mathfrak{H}\_t \right) \Delta t + \sigma(t\_i) \mathfrak{f}\_{t\_i}^{\frac{-(\ell-2)}{2}} \sqrt{\Delta} \mathcal{W}\_{i+1} \tag{30}$$

where *W*1, *W*2, ... , *WN* are *N* independent standard normal random variables. In this validation, the MC simulations based on the EM method (30) were conducted by MATLAB R2021a software on a quadcore Intel Core i5-1035G1 with 8 GB RAM.

**Example 5.** *In this example, we apply the MC simulations based on the CEV process [15]:*

$$\mathbf{d}r\_{t} = \kappa \left( \frac{\sigma\_{0}^{2} d \mathbf{e}^{2r\_{1}t}}{4\kappa} r\_{t}^{-(\ell-1)} - r\_{t} \right) \mathbf{d}t + \sigma\_{0} \mathbf{e}^{\sigma\_{1}t} r\_{t}^{-\ell \frac{\ell-2}{2}} \mathbf{d}W\_{t} \tag{31}$$

*where κ and σ*0 *are positive constants, σ*1 *is a non-negative constant and d is a positive integer greater than* 2*. By considering* (31) *and* (2)*, the parameter functions for SDE* (31) *are κ*(*t*) = *κ*,

*θ*(*t*) = *<sup>d</sup>σ*20 e2*σ*1*<sup>t</sup>*/4*κ and σ*(*t*) = *<sup>σ</sup>*0e*σ*1*t. Note that Assumptions 1 and 2 hold for these parameter functions. By Theorem 2, we have that:*

$$\mu\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \mathbf{e}^{-\gamma \kappa \tau} \sum\_{k=0}^{\infty} \mathfrak{f}\_k \tag{32}$$

*where:*

$$\mathfrak{F}\_k := \frac{1}{k!} \left( \prod\_{j=0}^{k-1} (\gamma - \ell j)(d + 2(\gamma - \ell j - 1)) \right) \left( \frac{\sigma\_0^2 e^{2\sigma\_1(T-\tau)} \left( \mathfrak{e}^{2\sigma\_1 \tau + \kappa \tau \ell} - 1 \right)}{4(2\sigma\_1 + \kappa \ell)} \right)^k r^{\gamma - \ell k}. \tag{33}$$

*However, Formula* (32) *can be reduced to a finite sum for a particular situation. By Corollary 1, if γ*/ ∈ Z+*, then:*

$$u\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \mathbf{e}^{-\gamma \kappa \tau} \sum\_{k=0}^{\gamma \prime \ell} \mathfrak{J}\_k. \tag{34}$$

*By Corollary 2, if there exists m* ∈ <sup>Z</sup>+0 *such that γ* = 1 − <sup>2</sup>*κ*(*τ*)*θ*(*τ*)/*σ*<sup>2</sup>(*τ*) + *m, which is* 1 − *d*/2 + *m in this example, for all τ* ∈ (0, *<sup>T</sup>*]*, then:*

$$u\_{\ell}^{\langle \gamma \rangle}(r, \tau) = \mathbf{e}^{-\gamma \kappa \tau} \sum\_{k=0}^{m} \xi\_k. \tag{35}$$

*Our experiments are classified into three cases: (i) γ*/ ∈ Z+*, (ii)* (*γ* − 1 + *d*/2)/ ∈ <sup>Z</sup>+0 *, and (iii) γ*/ ∈/ Z<sup>+</sup> *and* (*γ* − 1 + *d*/2)/ ∈/ <sup>Z</sup>+0 *. The algorithm of our validation is given in Algorithm 1. The parameters* = 2/3*, σ*0 = 0.01, *σ*1 = 0.02*, κ* = 0.03 *and T* = 10 *in the process* (31) *are set for all of these three cases. MC simulations were performed at each initial value r* = 0.1, 0.2, . . . , 2 *and τ* = 1, 2, . . . , 10*.*


7: Plot the resulting values and compare them with the surface of *<sup>u</sup>*(*<sup>r</sup>*, *τ*)

*For the case when γ*/ ∈ Z+*, we set d* = 3 *and consider two different values of γ. Here, we choose γ* = 2 *and* 8/3*. Figure 1 shows the comparison between Formula* (34) *and MC simulations. The results from MC simulations are presented by blue star markers, and Formula* (34) *is presented by the solid surfaces. All markers perfectly match with the surfaces. This indicates that our formula from Corollary 1 is correct. The validation runtimes for γ* = 2 *and* 8/3 *were* 23.82 *and* 22.30 *s, respectively.*

*For the case when* (*γ* − 1 + *d*/2)/ ∈ <sup>Z</sup>+0 *, we set d* = 4 *and consider γ* = 1 *and* 5/3*. Figure 2 demonstrates the comparison between Formula* (35) *and MC simulations. Evidently, the results from MC simulations and the surfaces from Formula* (35) *are completely coincident. Validation runtimes for γ* = 1 *and* 5/3 *were 22.34 and 22.63 s, respectively.*

(**a**) *γ* 2 (**b**) *γ* 83 **Figure 1.** The validation of conditional moments for process (31) where = 2/3, *σ*0 = 0.01, *σ*1 = 0.02, *κ*= 0.03,*T* = 10and *d*= 3withMCsimulations.

=

**Figure 2.** The validation of conditional moments for process (31) where = 2/3, *σ*0 = 0.01, *σ*1 = 0.02, *κ* = 0.03, *T* = 10 and *d* = 4 with MC simulations.

*For the case when γ*/ ∈/ Z<sup>+</sup> *and* (*γ* − 1 + *d*/2)/ ∈/ <sup>Z</sup>+0 *, we set d* = 5 *and consider γ* = 1*. Observe that from* (33)*,* |*ξ<sup>k</sup>*+1/*ξk*| *is* O(*k*) *as k* → ∞*; thus,* lim*k*→∞|*ξ<sup>k</sup>*+1/*ξk*| = ∞ *for* (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*]*. By the ratio test, the summation* ∑∞*<sup>k</sup>*=<sup>0</sup> *ξk diverges; hence, Formula* (32) *diverges for all* (*r*, *τ*) ∈ (0, ∞) × (0, *<sup>T</sup>*]*. This means that the conditional moment cannot be expressed in the form* (5)*. However, our experiment shows that finite terms of the summation in Formula* (32) *can be used to approximate the conditional moment. Figure 3 shows the comparison between the formula*

$$S\_n(r, \tau) := \mathbf{e}^{-\gamma \kappa \tau} \sum\_{k=0}^n \xi\_k^{\tau} \tag{36}$$

=

*for n* = 10, 1000 *and MC simulations. The results from MC simulations coincide with the surface from Formula* (36) *with n* = 10*. For n* = 1000*, the results from Formula* (36) *could not be computed by our machine. This supports our theory that Formula* (32) *diverges. Validation runtimes for n* = 10 *and n* = 1000 *were* 22.76 *and* 26.98 *s, respectively.*

**Figure 3.** The validation of conditional moments for process (31) where = 2/3, *γ* = 1, *σ*0 = 0.01, *σ*1= 0.02, *κ* = 0.03, *T* = 10 and *d* = 5 with MC simulations.

The next example shows a similar result of the third case in Example 5 for the IND-CEV process with constant parameter functions.

**Example 6.** *For SDE* (3) *with* = 2/3*, κ* = 0.03*, θ* = 0.003*, σ* = 0.01*, γ* = 1 *and T* = 10*, we have that γ*/ ∈/ Z<sup>+</sup> *and* (*γ* − 1 + <sup>2</sup>*κθ*/*σ*<sup>2</sup>)/ ∈/ <sup>Z</sup>+0 *. From Corollary 5, u*<sup>1</sup> 2/3(*<sup>r</sup>*, *τ*) *cannot be expressed in the form* (5)*. However, our experiment shows that finite terms of the summation in Formula* (15) *can be used to approximate the conditional moment. Let:*

$$\overline{S}\_{n}(r,\tau) := \sum\_{k=0}^{n} \frac{\mathbf{e}^{-\kappa\tau}}{k!} \left(\frac{\mathbf{e}^{2\kappa\tau} - 1}{2\kappa}\right)^{k} \left(\prod\_{j=0}^{k-1} \left(1 - 2j\right) \left(\kappa\theta - j\sigma^{2}\right)\right) r^{1-2k}.\tag{37}$$

*Figure 4 shows the comparison for Formula* (37) *between n* = 10, 1000 *and MC simulations. All blue markers match with the surface from the formula with n* = 10*, even though S* 5 *<sup>n</sup>*(*<sup>r</sup>*, *τ*) *diverges as n* → ∞*. Validation runtimes for n* = 10 *and n* = 1000 *were* 22.79 *and* 26.96 *s, respectively.*

**Figure 4.** The validation of conditional moments for process (3) where = 2/3, *γ* = 1, *κ* = 0.03, *θ* = 0.003 and *σ* = 0.01 and *T* = 10 with MC simulations.

 *n* =

### **7. Conclusions, Limitations and Future Researches**

 *n* =

In this study, we focused on the IND-CEV process (2) and a special case when the parameter functions are constants, which leads to process (3). We gave the sufficient conditions for SDE (2) in order to have a unique positive path-wise strong solution. We have derived the explicit formulas of conditional moments for this process. The derived formula for process (2) is shown in Theorem 2 in terms of infinite series. The formula can be reduced from infinite sum to finite sum for two situations: (i) the case when *γ*/ ∈ Z+, and (ii) condition (13), which are shown in Corollaries 1 and 2. Furthermore, we have presented the formula for process (3), where the parameter functions are constant, in Theorem 3. As a consequence, formulas for special situations are expressed in Corollaries 3 and 4. The characterization for the convergence of the infinite sum in the formula for process (3) is discussed in Theorem 4 and summarized in Corollary 5.

The use of our results was illustrated. This includes conditional moments, conditional variance and central moments, conditional mixed moments, conditional covariance and correlation. In addition, the moments of the stationary distribution of process (3) were proposed in Theorem 5.

Moreover, we have validated our closed-form formulas for process (2) by comparing the calculated values of conditional moments from our formula with the MC simulations via a number of experimental examples in Section 6. Our results in each situation have completely matched with MC simulations. Moreover, for some moments *γ* whose formula cannot be reduced to a finite sum, we can approximate the conditional moments by displaying the numerical result of the finite sum with suitable order. It turns out that the obtained results have good accuracy when compared with the MC simulations.

One major concern is that our proposed formulas in Theorem 2 and Corollaries 1 and 2 are not in closed form when integral terms cannot be analytically computed. In this case, a numerical method can be applied to calculate the coefficients numerically; see [28,29].

In the context of future works, our proposed closed-form formulas under the IND-CEV process have further beneficial aspects for pricing financial derivatives, such as moment swaps and the asset whose payoff can be generated by the conditional moments, see more details in [23,30]. In addition, since the transition PDF of process (2) is complicated and does not exist in closed form, our closed-form formulas can also be applied for parameter estimations of the behavior and dynamic of observed data; see more details in [9].

**Author Contributions:** Conceptualization, K.C., R.T. and C.T.; methodology, K.C., R.T. and C.T.; software, K.C., R.T. and C.T.; validation, K.C., R.T. and C.T.; formal analysis, K.C., R.T. and C.T.; investigation, K.C., R.T. and C.T.; writing—original draft preparation, K.C., R.T. and C.T.; writing— review and editing, K.C., R.T. and C.T.; visualization, K.C., R.T. and C.T.; supervision, K.C. and R.T.; project administration, K.C., R.T. and C.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We are grateful for a variety of valuable suggestions from the anonymous referees that have substantially improved the quality and presentation of the results. All errors are the authors' own responsibility. Thank the newborn son to make 24 June 2022 the best day of the first author's life.

**Conflicts of Interest:** The authors declare no conflict of interest.
