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Cambridge University Press, Journal of Applied Probability, 02(48), p. 420-434, 2011

DOI: 10.1017/s0021900200007968

Cambridge University Press, Journal of Applied Probability, 2(48), p. 420-434

DOI: 10.1239/jap/1308662636

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A lower bound for the first passage time density of the suprathreshold Ornstein-Uhlenbeck process

Journal article published in 2011 by Peter J. Thomas ORCID
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

We prove that the first passage time density $ρ(t)$ for an Ornstein-Uhlenbeck process $X(t)$ obeying $dX=-β X dt + σ dW$ to reach a fixed threshold $\theta$ from a suprathreshold initial condition $x_0>\theta>0$ has a lower bound of the form $ρ(t)>k \exp\left[-p e^{6β t}\right]$ 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 $β$, $σ$, $x_0$ and $\theta$, and discuss application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons. ; Comment: 15 pages, 1 figure