Asymptotic Results for First-Passage Times of Some Exponential Processes

We consider the process V (t) : t ≥ 0 defined by V (t) = v0eX(t) (for all t ≥ 0), where v0 > 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 V~ (t) : t≥ 0 , where V~ (t) = v0eX~(t) (for all t ≥ 0) and X~ (t) : t≥ 0 is the Normal approximation (as t→ ∞) of the process X(t) : t ≥ 0. In this paper we are interested in the first-passage times through a constant firing threshold β (where β > v0) for both processes V (t) : t ≥ 0 and V~ (t) : t≥ 0 ; our aim is to study their asymptotic behavior as β→ ∞ in the fashion of large deviations. We also study some statistical applications for both models, with some numerical evaluations and simulation results.