Applications of The Reflected Ornstein-Uhlenbeck Process

An Ornstein-Uhlenbeck process is the most basic mean-reversion model and has been used in various fields such as finance and biology. In some instances, reflecting boundary conditions are needed to restrict the state space of this process. We study an Ornstein-Uhlenbeck diffusion process with a reflecting boundary and its application to finance and neuroscience. In the financial application, the Vasicek model which is an Ornstein-Uhlenbeck process has been used to capture the stochastic movement of the short term interest rate in the market. The shortcoming of applying this model is that it allows a negative interest rate theoretically. Thus we use a reflected Ornstein-Uhlenbeck process as an interest rate model to get around this problem. Then we price zero-coupon bond and European options with respect to our model. In the application to neuroscience, we study integrate-and-fire (I-F) neuron models. We assume that the membrane voltage follows a reflected Ornstein-Uhlenbeck process and fires when it reaches a threshold. In this case, the interspike intervals (ISIs) are the same as the first hitting times of the process to a certain barrier. We find the first passage time density given ISIs using numerical inversion integration of the Laplace transform of the first passage time pdf. Then we estimate the unknown identifiable parameters in our model