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

Abstract

We prove that the first passage time density $ρ(t)$ for an Ornstein-Uhlenbeck process $X(t)$ obeying $dX=-βX dt + σdW$ to reach a fixed threshold $θ$ from a suprathreshold initial condition $x_0>θ>0$ has a lower bound of the form $ρ(t)>k \exp\left[-p e^{6βt}\right]$ 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 $β$, $σ$, $x_0$ and $θ$, and discuss application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.