**1. Introduction**

The stochastic differential equation (SDE) has been used to model various phenomena and investigate their properties, such as the moments, variance and conditional moments, which are beneficial for estimating parameters that play significant roles in several practical applications. For example, financial derivative prices, such as moment swaps, can be obtained by calculating the conditional moments of their payoffs under the risk neutral measure; see for more concrete studies Araneda et al. [1], Cao et al. [2], He and Zhu [3] and Nonsoong et al. [4]. Actually, such moments can be directly computed by employing SDE's transition probability density function (PDF). However, the transition PDF is often unavailable in closed form; so is the formula for those conditional moments of the SDE. Investigating properties of those SDEs is still imperative and challenging.

There are several empirical studies confirming that a mean-reverting drift process, such as the Vašíˇcek, Ornstein–Uhlenbeck (OU) [5] and Cox–Ingersoll–Ross (CIR) [6] processes, should not necessarily be linear. Indeed, the behaviors and dynamics of interest rate and its derivatives prefer nonlinearity in the mean-reverting drift rather than linear drift processes; see for more details in [7–10]. In order to extend the OU process, a nonlinear diffusion process was introduced by Cox [11], namely, the constant elasticity of variance (CEV) process. The CEV process is useful and has many applications in various fields. However, the drift term of Cox's CEV process is still linear. For many reasons described in the existing literature [7,8], an extended case of Cox's CEV process was first studied by Marsh

**Citation:** Chumpong, K.; Tanadkithirun, R.; Tantiwattanapaibul, C. Simple Closed-Form Formulas for Conditional Moments of Inhomogeneous Nonlinear Drift Constant Elasticity of Variance Process. *Symmetry* **2022**, *14*, 1345. https://doi.org/10.3390/ sym14071345

Academic Editor: Zhivorad Tomovski

Received: 1 June 2022 Accepted: 24 June 2022 Published: 29 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and Rosenfeld [12]. The process is sometimes called the Marsh–Rosenfeld (MR) process, and its transition PDF that can be straightforwardly calculated by using Itô's lemma and the transition PDF of the CIR process are very complicated; the closed-form formula for conditional moments of the MR process is also complicated or unavailable in general; see for more details in [13]. It gets even more complicated for an inhomogeneous-time MR process that extends the MR process by replacing the constant parameters in the process with time-dependent functions. From now on, we subsequently call the inhomogeneous-time MR process in general an inhomogeneous nonlinear drift constant elasticity of variance (IND-CEV) process.

Conditional moments have been extensively used in modern financial markets. For example, they can be used to price moment swaps. Unfortunately, the conditional moments, which can be directly computed by applying the transition PDFs, are often unavailable in closed form because the transition PDFs are hardly known. The Feynman–Kac technique is used to overcome this problem for calculating the conditional moments of many stochastic processes. There has still been little research on the analytical formula for conditional moments regarding the IND-CEV process. In this work, a novel approach is developed based on the Feynman–Kac theorem, where the partial differential equation (PDE) is solved analytically, and some combinatorial techniques are used to simplify the system of recursive ordinary differential equations (ODEs) associated with the conditional moment.

The rest of the paper is organized as follows. Section 2 provides an overview of the IND-CEV process and sufficient conditions of the time-dependent parameter functions in the process. The key methodology and main results are given in Section 3. Section 4 proposes some essential properties such as conditional moments, conditional variance and central moments, conditional mixed moments, conditional covariance and correlation. Section 5 provides the formula of the unconditional moments of the IND-CEV process with constant parameters. Experimental validations for our results applied with Monte Carlo (MC) simulations are addressed in Section 6. Conclusions, limitations and future researches are discussed in Section 7.
