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Thomas PJ. A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process. J Appl Probab 2016. [DOI: 10.1239/jap/1308662636] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck process X(t) obeying dX = -β Xdt + σdW to reach a fixed threshold θ from a suprathreshold initial condition x0 > θ > 0 has a lower bound of the form ρ(t) > kexp[-pe6βt] 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, x0, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.
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Di Crescenzo A, Giorno V, Nobile AG. On some first-crossing-time probabilities for a two-dimensional random walk with correlated components. ADV APPL PROBAB 2016. [DOI: 10.2307/1427699] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
For a two-dimensional random walk {X (n) = (X(n)1, X(n)2)T, n ∈ ℕ0} with correlated components the first-crossing-time probability problem through unit-slope straight lines x2 = x1 - r(r = 0,1) is analysed. The p.g.f.'s for the first-crossing-time probabilities are expressed as solutions of a fourth-degree algebraic equation and are then exploited to obtain the first-crossing-time probabilities. Several additional results, including the mean first-crossing time and the probability of ultimate crossing, are also given.
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