Moment Approximation of a Money Return Model Employing a Birth and Death Diffusion Process with General External Effect

Abstract

This paper derives moment approximation as well as the mean and the variance of a money return model within an important diffusion return process by considering the stochastic analogs of classical differences and differential equations. This is accomplished by employing an interest rate process that follows a birth and death diffusion process with general external effect process and represented as a solution to a stochastic differential equation. The analysis introduces a generalization of a widely applied statistical distribution, the exponential distribution. In particular, the moment approximation for some external effect distributions of Beta and Exponential distributions, as well as for the case of no external effects are generated. Numerical examples for a sample path of such a money return process are also considered for the case of fixed annualized interest rate and for the case of no jumps as well as the case of the occurrence of jump process that follow a uniform and exponential distribution. The results are useful in studying the behavior of the process and in statistical inference problems. The model generalization should attract wider applicability.