Abstract

This paper presentsãnãpplication of solutions of linear difference equations for obtainingã closed-form formula for the γ-th conditional moment of the Ornstein-Uhlenbeck (O-U) process, forãny positive real number γ. The partial differential equationãssociated with the O-U process is reduced toã system of ordinary differential equations, which can be solvedãnalytically in Laplace-transformed space using solutions of linear difference equations. Our success in performing Laplace inverse transform leads toã simple closed-form formula for the conditional moment. Interestingly, severalãsymptotic properties of the conditional moment can easily be deduced using our closed-form formula. Secondly, the n-th conditional moment of the trending O-U process is derived in closed form, forãny positive integer n. Finally, we derive the n-th unconditional moment of the O-U processãnd explore someãsymptotic properties.