We consider the first-passage time problem for the Feller-type diffusion process, having infinitesimal drift B1(x, t) = α(t) x + β(t) and infinitesimal variance B2(x, t) = 2 r(t)x, defined in the space state [0, +∞), with α(t) ∈ R, β(t) > 0, r(t) > 0 continuous functions. For the time- homogeneous case, some relations between the first-passage time densities of the Feller process and of the Wiener and the Ornstein–Uhlenbeck processes are discussed. The asymptotic behavior of the first-passage time density through a time-dependent boundary is analyzed for an asymptotically constant boundary and for an asymptotically periodic boundary. Furthermore, when β(t) = ξ r(t), with ξ > 0, we discuss the asymptotic behavior of the first-passage density and we obtain some closed-form results for special time-varying boundaries.

On the First-Passage Time Problem for a Feller-Type Diffusion Process

Virginia Giorno
;
Amelia G. Nobile
2021

Abstract

We consider the first-passage time problem for the Feller-type diffusion process, having infinitesimal drift B1(x, t) = α(t) x + β(t) and infinitesimal variance B2(x, t) = 2 r(t)x, defined in the space state [0, +∞), with α(t) ∈ R, β(t) > 0, r(t) > 0 continuous functions. For the time- homogeneous case, some relations between the first-passage time densities of the Feller process and of the Wiener and the Ornstein–Uhlenbeck processes are discussed. The asymptotic behavior of the first-passage time density through a time-dependent boundary is analyzed for an asymptotically constant boundary and for an asymptotically periodic boundary. Furthermore, when β(t) = ξ r(t), with ξ > 0, we discuss the asymptotic behavior of the first-passage density and we obtain some closed-form results for special time-varying boundaries.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4770427
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