Research articles
ScienceAsia 49S (2023):ID 59-67 |doi:
10.2306/scienceasia1513-1874.2023.s002
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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Center of Excellence in Data Science for Health Study, Division of Mathematics and Statistics, School of Science,
Walailak University, Nakhon Si Thammarat 80161 Thailand |
* Corresponding author, E-mail: rsanae@wu.ac.th
Received 17 Jan 2023, Accepted 17 Jul 2023
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