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) = v0e ˜X(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 (whereβ > 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.
D'Onofrio, G., Macci, C., Pirozzi, E. (2018). Asymptotic results for first-passage times of some exponential processes. METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY, 20(4), 1453-1476.
Asymptotic results for first-passage times of some exponential processes
Macci C
;
2018-01-01
Abstract
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) = v0e ˜X(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 (whereβ > 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.