Asymptotic Results for First-Passage Times of Some Exponential Processes

摘要

We consider the process {V (t) : t 0} defined by V (t) = v(0)e(X(t)) (for all t 0), where v(0) 0 and {X(t) : t 0} is a compound Poisson process with exponentially distributed jumps and a negative drift. This process can be seen as the neuronal membrane potential in the stochastic model for the firing activity of a neuronal unit presented in Di Crescenzo and Martinucci (Math Biosci 209(2):547-563 2007). We also consider the process (for all t 0) and is the Normal approximation (as) of the process {X(t) : t 0}. In this paper we are interested in the first-passage times through a constant firing threshold (where v(0)) for both processes {V (t) : t 0} and in the fashion of large deviations. We also study some statistical applications for both models, with some numerical evaluations and simulation results.