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 } $\{\tilde {V}(t):t\geq 0\}$ , where V ~ ( t ) = v 0 e X ~ ( t ) $\tilde {V}(t)=v_{0}e^{\tilde {X}(t)}$ (for all t ≥ 0) and { X ~ ( t ) : t ≥ 0 } $\{\tilde {X}(t):t\geq 0\}$ is the Normal approximation (as t → ∞ $t\to \infty $ ) 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 } $\{\tilde {V}(t):t\geq 0\}$ ; our aim is to study their asymptotic behavior as β → ∞ $\beta \to \infty $ in the fashion of large deviations. We also study some statistical applications for both models, with some numerical evaluations and simulation results.