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

A time-inhomogeneous Feller-type diffusion process with linear infinitesimal drift α(t)x+β(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t)x+\beta (t)$$\end{document} 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 β(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (t)$$\end{document} and the noise intensity function r(t) are connected via the relation β(t)=ξr(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (t)=\xi \,r(t)$$\end{document}, with 0≤ξ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \xi <1$$\end{document}. Various numerical computations are performed to illustrate the effect of the parameters on the first-passage time density, by assuming that α(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t)$$\end{document}, β(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (t)$$\end{document} or both of these functions exhibit some kind of periodicity.

with A 1 (x 0 , t 0 ) and A 2 (x 0 , t 0 ) given in (1), to solve imposing the initial delta condition and the absorbing boundary condition in the zero-state: Furthermore, let be the random variable describing the first-passage time (FPT) through the zero-state starting from X (t 0 ) = x 0 > 0; we denote by We note that the FPT density g(0, t|x 0 , t 0 ) is not affected by the boundary condition on the zero-state, provided that it is attainable. The problem of determining FPT densities for the Feller-type diffusion process arises in a variety of fields, including neurobiology, population dynamics, queueing systems and mathematical finance (cf., for instance, Linetsky [30], Masoliver and Perelló [35], Buonocore et al. [36], D'Onofrio et al. [37], Giorno et al. [38,39], Albano e Giorno [40], Di Nardo and D'Onofrio [41]). For instance, in population dynamics g(0, t|x 0 , t 0 ) describes the extinction density, whereas in queueing systems represents the busy period density. Lavigne and Roques in [18] focus on the distribution of the extinction times of a population whose size is described by a time-inhomogeneous Feller-type diffusion process with infinitesimal drift A 1 (x, t) = α(t) x and infinitesimal variance A 2 (x, t) = σ 2 x, where α(t) is a continuous function and σ 2 is a positive constant.
The functions (2) and (7) are intimately related; indeed, one has: Relation (8) shows that the determination of g(0, t|x 0 , t 0 ) requires the explicit evaluation of the transition pdf f a (x, t|x 0 , t 0 ) in the presence of an absorbing boundary at the zero-state.

Plain of the Paper
The paper is organized in five sections and seven appendices in which the proofs of the main results are reported. In Sect. 2, for the time-inhomogeneous Feller-type diffusion process X (t), with infinitesimal moments (1), we give some preliminary results concerning the Laplace transform (according to x 0 ) of the transition pdf f a (x, t|x 0 , t 0 ) in the presence of an absorbing boundary in the zero-state. The proportional case, in which the immigration intensity function β(t) and the noise intensity function r (t) are related as β(t) = ξ r (t), with 0 ≤ ξ < 1, is also analyzed. In Sect. 3, the transition pdf f a (x, t|x 0 , t 0 ) is obtained for the process (1) in the general case, by distinguishing the case x = 0 (Sect. 3.1) and x > 0 (Sect. 3.2). In Sect. 4, we focus on the FPT of X (t) through the zero-state for the general case and we determine the expression of the FPT pdf g(0, t|x 0 , t 0 ). In Sects. 3 and 4, we also show as the results of the proportional case can be derived from the general case. In Sect. 5, various numerical computations are performed making use of MATHEMATICA to illustrate the effect of periodic intensity functions on the FPT pdf g(0, t|x 0 , t 0 ). Specifically, we assume that the growth intensity function α(t), the immigration intensity function β(t) or both these functions exhibit some kind of periodicity. The FPT mean t 1 (0, t|x 0 , t 0 ) and the coefficient of variation CV(0|x 0 , t 0 ) = √ Var(0|x 0 , t 0 )/t 1 (0|x 0 , t 0 ) are also analyzed.

Preliminary Results
In this section, we determine the Laplace transform (according to x 0 ) of the transition pdf f a (x, t|x 0 , t 0 ) in the general case. Furthermore, the explicit expressions of the transition pdf and of the FPT density through the zero-state are obtained in the proportional case.

Laplace Transform
For t ≥ t 0 and x ≥ 0, we consider the Laplace transform: We determine Z a (x, t|s, t 0 ) so that, by taking its inverse Laplace transform, we obtain f a (x, t|x 0 , t 0 ). Multiplying both sides of (3) by e −sx 0 , integrating with respect to x 0 over the interval [0, +∞) and making use of the boundary condition (5), we have the following partial differential equation to solve with the initial condition derived from (9) by using the initial condition (4).
For t ≥ t 0 , we have: where Proof The proof is given in Appendix A.

Proportional Case
For all t ≥ 0, we suppose that the continuous functions β(t) and r (t) are proportional, i.e.
Proposition 2 Under the assumption (14), for t ≥ t 0 one has: Furthermore, the transition pdf of X (t) in the presence of an absorbing boundary in the zero-state is: with A(t|t 0 ) and R(t|t 0 ) defined in (13) and where denotes the modified Bessel function of the first kind.
Proof The proof is given in Appendix B.
Note that, the first of (16) follows by taking the limit as x ↓ 0 in the second, recalling that for fixed ν and for z → 0 one has (cf. Abramowitz and Stegun [42], p. 375, no 9.6.7): If (14) holds, for t ≥ t 0 , x > 0 and x 0 > 0 from (16) it follows: Proposition 3 Under the assumption (14), for t ≥ t 0 and x 0 > 0 one has: with R(t|t 0 ) given in (13) and where γ (a, z) = z 0 e −y y a−1 dy, Re a > 0 ( 2 0 ) denotes the incomplete gamma function.

General Case
We assume that α(t), β(t) and r (t) are continuous functions such that where with A(t|t 0 ) and R(t|t 0 ) given in (13). We note that V a (x, t|s, t 0 ) does not dependent upon β(t). Therefore, to obtain the transition pdf f a (x, t|x 0 , t 0 ) for X (t) with infinitesimal moments (1), we proceed as follows: (1) we determine the transition pdf f a (0, t|x 0 , t 0 ) for x 0 > 0 and t ≥ t 0 by taking the inverse Laplace transform of Z a (0, t|s, t 0 ); (2) we find the inverse Laplace transform v a (x, t|x 0 , t 0 ) of (29) and we calculate the transition pdf f a (x, t|x 0 , t 0 ) as a convolution, according to x 0 , between f a (0, t|x 0 , t 0 ) and the function v a (x, t|x 0 , t 0 ) for x > 0, x 0 > 0 and t ≥ t 0 .

General Case: x = 0
In this section, we obtain the transition pdf in the presence of an absorbing boundary in the zero-state when the process X (t) reaches x = 0 at time t ≥ t 0 . By setting x = 0 in (12), for t ≥ t 0 we obtain: with A(t|t 0 ) and R(t|t 0 ) defined in (13).
In the sequel, we denote by B n (d 1 , d 2 , . . . , d n ) the complete Bell polynomials, recursively defined as follows: Proposition 5 Under the assumption of Proposition 1, for t ≥ t 0 and x 0 > 0 the transition pdf of the time-inhomogeneous Feller-type diffusion process X (t) with an absorbing boundary in the zero-state is where with A(t|t 0 ) and R(t|t 0 ) defined in (13), B n (d 1 , d 2 , . . . , d n ) given in (31) and in (32), and denoting the Laguerre polynomials.
Proof The proof is given in Appendix C.

General Case: x > 0
In this section, we obtain the transition pdf Proposition 6 Under the assumption of Proposition 1, for x 0 > 0 and t ≥ t 0 , one has: with A(t|t 0 ) and R(t|t 0 ) defined in (13), whereas δ(x) denotes the delta Dirac function and I ν (z) represents the Bessel function modified of first kind.
Proof The proof is given in Appendix D. (40) is the sum of two terms. The second term in (40) identifies For x > 0, the transition pdf f a (x, t|x 0 , t 0 ) can be obtained via a convolution, according to x 0 , between the pdf f a (0, t|x 0 , t 0 ) and the function v a (x, t|x 0 , t 0 ), determined in Propositions 5 and 6, respectively: Proposition 7 Under the assumption of Proposition 1, for t ≥ t 0 , x > 0 and x 0 > 0 one has: with A(t|t 0 ) and R(t|t 0 ) given in (13) and Ψ (t|z, t 0 ) defined in (34).
Note that, by taking the limit as x ↓ 0 in (42), we obtain (33).

The First-Passage Time Through the Zero-State
We now focus on the distribution function of the FPT through the zero-state for the timeinhomogeneous Feller-type diffusion process X (t), with infinitesimal moments (1), when α(t), β(t) and r (t) are continuous functions such that α(t) ∈ R, β(t) ∈ R, r (t) > 0, β(t) ≤ ξ r (t), with 0 ≤ ξ < 1. The FPT problem of X (t) through the zero-state can be studied starting from Eq. (8) and making use of (42).
Proof The proof is given in Appendix E.

Proof
The proof is given in Appendix F.

Special Cases
Under the assumption (14), we analyze the cases in which the growth intensity function α(t), or the immigration intensity function β(t) or both of them have some kind of periodicity. These cases are of interest in various applied fields, such as in population growth and in queueing systems. Indeed, periodic immigration intensity functions play an important role in the description of the evolution of dynamic for systems influenced by seasonal immigration or other regular environmental cycles. Furthermore, periodic growth intensity functions express the existence of fluctuation in the population dynamics and the presence of rush hours occurring on a daily basis in queueing systems.

Periodic Immigration Intensity Function
We consider the time-inhomogeneous Feller-type process X (t) such that with α ∈ R, 0 ≤ ξ < 1 and where ν > 0 is the average of the periodic function r (t) of period Q, c is the amplitude of the oscillations, with 0 ≤ c < 1. From (13), for t ≥ t 0 one has A(t|t 0 ) = α (t − t 0 ) and (55) Then, from (55) one obtains: so that, by virtue of (23), the FPT through the zero-state is a certain event for α ≤ 0. Moreover, for α = 0 the FPT moments (26) are divergent. In Figs. 1, 2 and 3, the FPT distribution G(0, t|x 0 , t 0 ) = 1 − +∞ 0 f a (x, t|x 0 , t 0 ) dx, obtained making use of (19), and the FPT pdf g(0, t|x 0 , t 0 ), given in (22), are plotted as function of t for the diffusion process (53) for some choices of parameters. In Fig. 4, the mean t 1 (0|x 0 , t 0 ) and the coefficient of variation CV(0|x 0 , t 0 ), obtained making use of (26), are plotted as function of ν for ξ = 0, 0.3, 0.6. We note that as ν increases, the FPT mean t 1 (0|x 0 , t 0 ) decreases whereas the coefficient of variation increases. Instead, as ξ increases in [0, 1), the FPT mean increases and the coefficient of variation decreases, due to a raise of the immigration intensity function.

Periodic Growth Intensity Function
We consider the time-inhomogeneous Feller-type process X (t) such that with r > 0, 0 ≤ ξ < 1 and where η ∈ R is the average of the periodic function α(t) of period Q 1 , b determines the amplitude of the oscillations, with 0 ≤ b < 1. In Fig. 5, the intensity function (57) is plotted as function of t for some choices of parameters η, b and Q 1 . The dotted lines refer to the average cases, in which α(t) = η with η = −5 (bottom) and η = 5 (top). From (13), for t ≥ t 0 one has and (59) Fig. 5 The intensity function α(t), given in (57), is plotted as function of t for some choices of parameters. The dotted lines refer to the average cases Then, from (59) one obtains: so that, by virtue of (23), the FPT through zero-state is a certain event for η ≤ 0. Moreover, for η = 0 the FPT moments (26) are divergent.
In Fig. 6, the FPT pdf g(0, t|x 0 , t 0 ), given in (22), is plotted as function of t for the process (56) for some choices of parameters. Instead, in Fig. 7, the mean t 1 (0|x 0 , t 0 ) and the coefficient of variation CV(0|x 0 , t 0 ), obtained making use of (26), are plotted as function of r for ξ = 0, 0.3, 0.6. We note that as r increases, the FPT mean t 1 (0|x 0 , t 0 ) decreases, whereas the coefficient of variation increases. Moreover, the FPT mean and the coefficient of variation increase with ξ in [0, 1).

Periodic Immigration and Growth Intensity Functions
We consider the time-inhomogeneous Feller-type process X (t) such that with 0 ≤ ξ < 1, r (t) defined in (54) and α(t) given in (57). Recalling (13), for t ≥ t 0 one obtains A(t|t 0 ) given in (58) and The explicit expression of R(t|t 0 ) in (61) is obtained in Appendix G. We note that lim t→+∞ R(t|t 0 ) diverges as η ≤ 0, so that, due to (23), the FPT through the zero-state is a certain event for X (t).
In Fig. 8, the FPT pdf g(0, t|x 0 , t 0 ), given in (22), is plotted as function of t for the process (60) for some choices of parameters. Comparing Figs. 6 and 8 , we note the effect of the different periodicities of the growth intensity function α(t), with Q 1 = 1, and of the immigration intensity function β(t) = ξ r (t), with Q = 2. In Fig. 9, the mean t 1 (0|x 0 , t 0 ) and the coefficient of variation CV(0|x 0 , t 0 ), obtained making use of (26), are plotted as function of ν for ξ = 0, 0.3, 0.6. As ν increases, the FPT mean t 1 (0|x 0 , t 0 ) decreases whereas the coefficient of variation increases. Instead, as ξ increases in [0, 1), both the FPT mean and the coefficient of variation increase.

Concluding Remarks
In this paper, we have considered a time-inhomogeneous Feller-type diffusion process We have assumed that the zero-state represents an absorbing boundary for X (t). This process plays a relevant role in different fields, including physics, biology, neuroscience, finance and others. For instance, in population biology α(t) represents the growth intensity function and can be positive, negative or zero at different time instants, β(t) describes the immigration intensity function; instead, the noise intensity function r (t) takes into account the environmental fluctuations. For this process, the transition density f a (x, t|x 0 , t 0 ) in the presence of an absorbing boundary in zero-state and the FPT density g(0, t|x 0 , t 0 ) from X (t 0 ) = x 0 to the zero-state are obtained. Special attention is dedicated to the proportional case, in which the immigration intensity function and the noise intensity function are related as β(t) = ξ r (t), with 0 ≤ ξ < 1. Various numerical computation are performed to illustrate the effect of periodic intensity functions on the FPT pdf g(0, t|x 0 , t 0 ), by assuming that α(t), β(t) or both these functions exhibit some kind of periodicity.
with the initial conditions: so that i.e. the condition (5) is satisfied.