A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process

Abstract

We prove that the first passage time density $\rho(t)$ for an Ornstein-Uhlenbeck process $X(t)$ obeying $dX=-\beta X dt + \sigma dW$ to reach a fixed threshold $\theta$ from a suprathreshold initial condition $x_0>\theta>0$ has a lower bound of the form $\rho(t)>k \exp\left[-p e^{6\beta t}\right]$ for positive constants $k$ and $p$ for times $t$ exceeding some positive value $u$. We obtain explicit expressions for $k, p$ and $u$ in terms of $\beta$, $\sigma$, $x_0$ and $\theta$, and discuss application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.