*Article* **Simple Closed-Form Formulas for Conditional Moments of Inhomogeneous Nonlinear Drift Constant Elasticity of Variance Process**

**Kittisak Chumpong 1,2, Raywat Tanadkithirun 3,\* and Chanon Tantiwattanapaibul 3**


**Abstract:** The stochastic differential equation (SDE) has been used to model various phenomena and investigate their properties. Conditional moments of stochastic processes can be used to price financial derivatives whose payoffs depend on conditional moments of underlying assets. In general, the transition probability density function (PDF) of a stochastic process is often unavailable in closed form. Thus, the conditional moments, which can be directly computed by applying the transition PDFs, may be unavailable in closed form. In this work, we studied an inhomogeneous nonlinear drift constant elasticity of variance (IND-CEV) process, which is a class of diffusions that have time-dependent parameter functions; therefore, their sample paths are asymmetric. The closed-form formulas for conditional moments of the IND-CEV process were derived without having a condition on eigenfunctions or the transition PDF. The analytical results were examined through Monte Carlo simulations.

**Keywords:** conditional moment; constant elasticity of variance process; Feynman–Kac formula

**MSC:** 34A30; 60G65; 62M20; 65C05
