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)=2r(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)=ξr(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.
Highlights
Dipartimento di Informatica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 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
We focus our analysis on the asymptotic behavior of the first-passage time” (FPT) pdf for the Feller diffusion process, with α < 0, β > 0 and r > 0, by considering separately two cases: S(t) is an asymptotically constant boundary and S(t) is an asymptotically periodic boundary
We focus on the asymptotic behavior of the FPT pdf of the Feller-type diffusion process (69), with a zero-flux condition in the zero state, through the asymptotically constant boundary (61), with S(t) > 0, where η (t) ∈ C1 [t0, +∞) is a bounded function, that does not depend on S, such that (62) holds
Summary
We consider the time-homogeneous Feller process X (t) with drift B1 ( x ) = αx + β and infinitesimal variance B2 ( x ) = 2 r x, defined in the state space [0, +∞). As proved by Feller [44], the state x = 0 is an exit boundary for β ≤ 0, a regular boundary for 0 < β < r and an entrance boundary for β ≥ r. The scale function and the speed density of X (t) are (cf Karlin and Taylor [45]):. We assume that β > 0 and suppose that a zero-flux condition is placed in the zero state
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