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An application of solutions of linear difference equations for obtaining the conditional moments of the trending Ornstein-Uhlenbeck processes

Nopporn Thamrongrat, Kong Kanjanasopon, Sanae Rujivan^*

 
ABSTRACT:     This paper presents an application of solutions of linear difference equations for obtaining a closed-form formula for the ?-th conditional moment of the Ornstein-Uhlenbeck (O-U) process, for any positive real number ?. The partial differential equation associated with the O-U process is reduced to a system of ordinary differential equations, which can be solved analytically in Laplace-transformed space using solutions of linear difference equations. Our success in performing Laplace inverse transform leads to a simple closed-form formula for the conditional moment. Interestingly, several asymptotic 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, for any positive integer n. Finally, we derive the n-th unconditional moment of the O-U process and explore some asymptotic properties.

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* Corresponding author, E-mail: rsanae@wu.ac.th

Received 17 Jan 2023, Accepted 17 Jul 2023