Time-Inhomogeneous Feller-type Diffusion Process with Absorbing Boundary Condition

Abstract

A time-inhomogeneous Feller-type diffusion process with linear infinitesimal drift $$\alpha (t)x+\beta (t)$$ α ( t ) x + β ( t ) and linear infinitesimal variance 2r(t)x is considered. For this process, the transition density in the presence of an absorbing boundary in the zero-state and the first-passage time density through the zero-state are obtained. Special attention is dedicated to the proportional case, in which the immigration intensity function $$\beta (t)$$ β ( t ) and the noise intensity function r(t) are connected via the relation $$\beta (t)=\xi \,r(t)$$ β ( t ) = ξ r ( t ) , with $$0\le \xi <1$$ 0 ≤ ξ < 1 . Various numerical computations are performed to illustrate the effect of the parameters on the first-passage time density, by assuming that $$\alpha (t)$$ α ( t ) , $$\beta (t)$$ β ( t ) or both of these functions exhibit some kind of periodicity.

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

“…We assume that α(t) ∈ R, r(t) > 0 and β(t) = ξ r(t), with 0 ≤ ξ < 1, in (49). As proven in Giorno and Nobile [45] one has…”

“…In the sequel, we assume that α ∈ R, β ∈ R, r > 0, with β < r, and an absorbing condition is set in the zero-state. As proven in Giorno and Nobile [45], for a TH-F process Z(t) having β ∈ R, r > 0, with β < r, one has…”

“…We point out that the general TNH-F process with an absorbing boundary in zero is considered in Giorno and Nobile [45], Masoliver and Perelló [46], Masoliver [47] and Lavigne and Roques [48].…”

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

“…Note that a general expression of the FPT density for the time-inhomogeneous Fellertype process (1) through the zero state is given by Giorno and Nobile [43].…”

“…The renewal Equation (3) expresses that any sample path that reaches x ≥ S(t) [x ≤ S(t)], after starting from x 0 < S(t 0 ) [x 0 > S(t 0 )] at time t 0 , must necessarily cross S(u) for the first time at some intermediate instant u ∈ (t 0 , t). Research on the FPT problem for the Feller diffusion process has been carried out by Giorno et al [37], Linetsky [38], Masoliver and Perelló [39], Masoliver [40], Chou and Lin [41], Di Nardo and D'Onofrio [42], Giorno and Nobile [43]).…”