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

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.

On the Absorbing Problems for Wiener, Ornstein–Uhlenbeck, and Feller Diffusion Processes: Similarities and Differences

“…For diffusion processes, closed form expressions for FPT densities through constant boundaries are not available, except in some special cases (see Ricciardi et al [12], Ding and Rangarajan [13], Molini et al [14], Giorno and Nobile [15], Masoliver [16]). In particular, closed form expressions are available in the following cases: (i) the Wiener process through an arbitrary constant boundary; (ii) the Ornstein-Uhlenbeck process through the boundary in which the drift vanishes; and (iii) the Feller process through the zero state.…”