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Stability, thermophsical properties of nanofluids, and applications in solar collectors: A review

  • Recently, renewable energies have attracted the significant attention of scientists. Nanofluids are fluids carrying nano-sized particles dispersed in base fluids. The improved heat transfer by nanofluids has been used in several heat-transfer applications. Nanofluids' stability is very essential to keep their thermophysical properties over a long period of time after their production. Therefore, a global approach including stability and thermophysical properties is necessary to achieve the synthesis of nanofluids with exceptional thermal properties. In this context, the objective of this paper is to summarize current advances in the study of nanofluids, such as manufacturing procedures, the mechanism of stability assessment, stability enhancement procedures, thermophysical properties, and characterization of nanofluids. Also, the factors influencing thermophysical properties were studied. In conclusion, we discuss the application of nanofluids in solar collectors.

    Citation: Omar Ouabouch, Mounir Kriraa, Mohamed Lamsaadi. Stability, thermophsical properties of nanofluids, and applications in solar collectors: A review[J]. AIMS Materials Science, 2021, 8(4): 659-684. doi: 10.3934/matersci.2021040

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  • Recently, renewable energies have attracted the significant attention of scientists. Nanofluids are fluids carrying nano-sized particles dispersed in base fluids. The improved heat transfer by nanofluids has been used in several heat-transfer applications. Nanofluids' stability is very essential to keep their thermophysical properties over a long period of time after their production. Therefore, a global approach including stability and thermophysical properties is necessary to achieve the synthesis of nanofluids with exceptional thermal properties. In this context, the objective of this paper is to summarize current advances in the study of nanofluids, such as manufacturing procedures, the mechanism of stability assessment, stability enhancement procedures, thermophysical properties, and characterization of nanofluids. Also, the factors influencing thermophysical properties were studied. In conclusion, we discuss the application of nanofluids in solar collectors.



    Reflected diffusion processes are used extensively in various applied fields, such as mathematical biology, queueing theory, finance and neuroscience. In particular, diffusion processes with a reflection condition at the origin arise as diffusion limits of a number of classical birth-death processes in population dynamics and as a heavy-traffic approximation for queueing systems (see, for instance, Giorno et al. [1], Ward and Glynn [2], Di Crescenzo et al. [3,4]). Reflected diffusion processes are also applied in economics and finance for modeling regulated markets, interest rates and stochastic volatility (cf., for instance, Linetsky [5], Veestraeten [6]). Moreover, in neuronal models, the membrane potential evolution can be described by focusing the attention on the diffusion processes confined by a lower reflecting boundary that can be interpreted as the neuronal reversal hyperpolarization potential (cf. Lánský and Ditlevsen [7], Buonocore et al. [8,9], D'Onofrio et al. [10]). References to other applications of reflected diffusion processes in neuroscience, in population dynamics, in economics, in finance and in queueing systems can be found in Di Crescenzo et al. [11], Giorno and Nobile [12,13], Mishura and Yurchenko-Tytarenko [14]).

    In various types of instances, first-passage time (FPT) problems are invoked to describe events such as extinction in population dynamics, busy period in queueing systems and firing times in neuronal modeling (cf., for instance, Ricciardi et a. [15], Masoliver and Perelló [16], Bo et al. [17], Abundo and Pirozzi [18], Giorno and Nobile [19]).

    In several applications, it is useful to consider diffusion processes with linear infinitesimal drift and linear infinitesimal variance, having state-space [0,+) with a zero-flux condition at the zero-state. This class incorporates the Wiener, Ornstein-Uhlenbeck and Feller diffusion processes with reflection in the zero-state. Such processes are widely used in the modeling queueing systems under assumptions of heavy-traffic and in the description of populations growth in a random environment. In these contexts, the number of customers or individuals is bound to take non negative values, so that a reflection condition in the zero-state must thus be imposed. In particular, in queueing systems a reflecting condition in the zero-state is required because new customers can access the system if it is empty; moreover, in the population dynamics the reflection in zero allows to include immigration effects. In other contexts, the reflecting boundary can refer to a non-zero state and can be also time-dependent (cf., for instance, [8,9]). In the present paper, we restrict the attention to the Wiener, Ornstein-Uhlenbeck and Feller diffusion processes in [0,+) with a reflection in the zero-state.

    In Section 2, we briefly review some background results on the time-inhomogeneous and time-homogeneous diffusion processes restricted to the interval [0,+), with a reflecting or a zero-flux condition in the zero-state. In this case, a time-inhomogeneous diffusion process is characterized by time-dependent infinitesimal moments, whereas the infinitesimal moments of a time-homogeneous diffusion process are not time-dependent.

    In Sections 3–5, we consider the time-inhomogeneous Wiener, Ornstein-Uhlenbeck and Feller diffusion processes in [0,+) (cf. Table 1).

    Table 1.  Time-inhomogeneous diffusion processes considered in Sections 3–5.
    Time-inhomogeneous process Infinitesimal drift and infinitesimal variance
    X(t) - Reflected Wiener (TNH-RW) A1(t)=β(t),A2(t)=σ2(t)(β(t)R,σ(t)>0)
    Y(t) - Reflected Ornstein-Uhlenbeck (TNH-ROU) B1(x,t)=α(t)x+β(t),B2(t)=σ2(t)(α(t)R,β(t)R,σ(t)>0)
    Z(t) - Reflected Feller (TNH-RF) C1(x,t)=α(t)x+β(t),C2(x,t)=2r(t)x(α(t)R,β(t)>0,r(t)>0)

     | Show Table
    DownLoad: CSV

    We determine closed form expressions for the transition probability density function (pdf) and for the related conditional moments of the first and the second order in the following cases:

    ● for the TNH-RW process X(t) with β(t)=γσ2(t), where γR;

    ● for the TNH-ROU process Y(t) with β(t)=γσ2(t)eA(t|0), where γR and A(t|0)=t0α(u)du;

    ● for the TNH-RF process Z(t) with β(t)=ξr(t), where ξ>0.

    For these processes, we analyze the asymptotic behavior of the transition densities. Moreover, in Section 5, for β(t)=r(t)/2 and for β(t)=3r(t)/2 some relationships between the transition pdf for Wiener, Ornstein-Uhlenbeck and Feller processes are proved.

    In Sections 3–5, we also take into account the time-homogeneous Wiener, Ornstein-Uhlenbeck and Feller diffusion processes in [0,+) (see Table 2). For these processes, we study:

    Table 2.  Time-homogeneous diffusion processes considered in Sections 3–5.
    Time-homogeneous process Infinitesimal drift and infinitesimal variance
    X(t) - Reflected Wiener (TH-RW) A1=β,A2=σ2(βR,σ>0)
    Y(t) - Reflected Ornstein-Uhlenbeck (TH-ROU) B1(x)=αx+β,B2=σ2(α0,βR,σ>0)
    Z(t) - Reflected Feller (TH-RF) C1(x)=αx+β,C2(x)=2rx(αR,β>0,r>0)

     | Show Table
    DownLoad: CSV

    ● the asymptotic transition pdf (steady-state density);

    ● the transition pdf for the TH-RW and for the TH-RF processes;

    ● the Laplace transform (LT) of the transition pdf (βR) and the transition pdf (β=0) for the TH-ROU process;

    ● the asymptotic behavior of the FPT moments for boundaries near to zero-state and for large boundaries.

    Moreover, the mean, the coefficient of variation and the skewness of FPT for the TH-RW, TH-ROU and TH-RF processes are analyzed for various choices of parameters making use of Mathematica.

    In Section 6, we consider some examples of the TNH-RW, TNH-ROU and TNH-RF diffusion processes useful to modeling queueing systems in heavy-traffic conditions; for these processes, we assume that the infinitesimal drift and the infinitesimal variance are time-dependent and include periodic functions. Several numerical computations with Mathematica are performed to analyze the conditional averages and the conditional variances and their asymptotic behaviors for some choices of the periodic functions and of the parameters.

    This paper is dedicated to the memory of Patricia Román Román. Her untimely death leaves a great hole in our scientific community and an even greater hole in our hearts.

    In this section, we briefly review some results on the diffusion processes that will be used in the next sections to analyze Wiener, Ornstein-Uhlenbeck and Feller diffusion processes in [0,+), with a zero-flux condition at the zero-state.

    Let D(t) be a time-inhomogeneous diffusion (TNH-RD) process with infinitesimal drift ζ1(x,t) and the infinitesimal variance ζ2(x,t), restricted to interval [0,+), with + unattainable end-point and a zero-flux condition in the zero-state. The transition pdf rD(x,t|x0,t0) of D(t) can be obtained as the solution of the Fokker-Planck equation (cf. Dynkin [20])

    rD(x,t|x0,t0)t=x{ζ1(x,t)rD(x,t|x0,t0)}+122x2{ζ2(x,t)rD(x,t|x0,t0)}, (2.1)

    with the initial delta condition limtt0rD(x,t|x0,t0)=δ(xx0) and the condition:

    limx0{ζ1(x,t)rD(x,t|x0,t0)12x[ζ2(x,t)rD(x,t|x0,t0)]}=0. (2.2)

    We note that (2.1) can be re-written as

    rD(x,t|x0,t0)t=jD(x,t|x0,t0)x,

    where

    jD(x,t|x0,t0)=ζ1(x,t)rD(x,t|x0,t0)12x{ζ2(x,t)rD(x,t|x0,t0)}

    represents probability flux (or current) of D(t). Eq. (2.2) is the zero-flux or reflecting condition in the zero-state and corresponds to requiring that +0rD(x,t|x0,t0)dx=1 for all tt0.

    Expressions in closed form for rD(x,t|x0,t0) can be obtained only for some choices of the infinitesimal moments. For instance, if D(t) is a TNH-RD process with space-state [0,+), having infinitesimal drift and infinitesimal variance

    ζ1(x,t)=dh1(t)+[xdh1(t)]h2(t)h2(t),ζ2(x,t)ζ2(t)=h1(t)h2(t)h1(t)h2(t), (2.3)

    with the "prime" symbol denoting derivative with respect to the argument, dR, h2(t)0 and h1(t)/h2(t) is a non-negative and monotonically increasing function, then rD(x,t|x0,t0) can be determined in closed form (cf. Buonocore et al. [8]):

    rD(x,t|x0,t0)=fD(x,t|x0,t0)x[exp{2dxh2(t)}FD(x,t|x0,t0)],x0,x00, (2.4)

    where fD(x,t|x0,t0) and FD(x,t|x0,t0)=xfD(z,t|x0,t0)dz are the transition pdf and probability distribution function of the corresponding unrestricted diffusion process with space-state R, respectively. Specifically, for the diffusion process (2.3), fD(x,t|x0,t0) is a normal density with mean and variance

    MD(t|x0,t0)=h2(t)h2(t0)x0+dh2(t0)[h1(t)h2(t0)h1(t0)h2(t)],VD(t|t0)=h2(t)h2(t0)[h1(t)h2(t0)h1(t0)h2(t)].

    We remark that rD(x,t|x0,t0), given in (2.4), satisfies the Fokker-Planck equation (2.1), the initial delta condition, the zero-flux condition (2.2) and also +0rD(x,t|x0,t0)dx=1. Moreover, under suitable conditions, the infinitesimal moments of the Wiener process and of the Ornstein-Uhlenbeck process satisfy (2.3); in these cases, the transition pdf rD(x,t|x0,t0) is obtainable from (2.4). Instead, for the Feller process the infinitesimal variance depends on x, so that (2.4) does not hold.

    The knowledge of density rD(x,t|x0,t0) allows to evaluate the conditional moments:

    E[Dk(t)|D(t0)=x0]=+0xkrD(x,t|x0,t0)dx,k=1,2, (2.5)

    For a time-homogeneous reflected diffusion (TH-RD) process D(t), with state-space [0,+), one has ζ1(x,t)=ζ1(x) and ζ2(x,t)=ζ2(x) for all t. For the TH-RD process, the transition pdf rD(x,t|x0,t0)=rD(x,tt0|x0,0)=rD(x,tt0|x0) and, in this case, we assume that t0=0.

    We denote by

    hD(x)=exp{2xζ1(u)ζ2(u)du},sD(x)=2ζ2(u)hD(u), (2.6)

    the scale function and the speed density. These functions allow to classify the end-points of a diffusion process into regular, natural, exit, entrance and in attracting or nonattracting boundaries (cf. Karlin and Taylor [21]). Let

    TD(S|x0)={inft0{t:D(t)S},0D(0)=x0<S,inft0{t:D(t)S},D(0)=x0>S0,

    be the random variable that describes the FPT of D(t) through S starting from D(0)=x0S. We denote by gD(S,t|x0)=dP{TD(S|x0)t}/dt the FPT density and by PD(S|x0)=+0gD(S,t|x0)dt the ultimate FPT probability. When PD(S|x0)=1, let

    t(D)n(S|x0)=+0tngD(S,t|x0)dt,n=1,2,

    be the n-th FPT moment. The functions (2.6) allow to determine the FPT moments thanks to the Siegert formula (cf. Siegert [22]). Specifically, if D(t) is a TH-RD process in [0,+), with + nonattracting end-point, for n=1,2, it results:

    ● If x0>S>0, one has PD(S|x0)=1 and, if +zsD(u)du converges, the FPT moments can be evaluated as:

    t(D)n(S|x0)=nx0SdzhD(z)+zsD(u)t(D)n1(S|u)du,x0>S>0. (2.7)

    Equation (2.7) holds also for S=0 provided that the zero-state is a regular boundary.

    ● If 0x0<S, one has PD(S|x0)=1 and the FPT moments can be iteratively computed as:

    t(D)n(S|x0)=nSx0dzhD(z)z0sD(u)t(D)n1(S|u)du,0x0<S. (2.8)

    Making use of (2.8), in Giorno et al. [23] the following asymptotic results for the FPT moments are proved when the boundary S moves indefinitely away from the zero-state and when S is in the neighborhood of zero.

    Remark 2.1. For the TH-RD process D(t), we denote by

    k1(x)=ζ2(x)hD(x)x0sD(u)du,k2(x)=[ζ1(x)ζ2(x)4]hD(x)x0sD(u)du.

    One has:

    1. If limx+k1(x)=+ and limx+k2(x)=, then

    limS+t(D)n(S|0)[t(D)1(S|0)]n=n!,n=0,1,2,

    2. If limx0k1(x)=0 and limx0k2(x)=ν, with <ν<1, then

    limS0t(D)n(S|0)[t(D)1(S|0)]n=un,n=0,1,2,

    where

    u0=1,un=nk=1(nk)(1)k1{k1i=0[1+2i(1ν)]}1unk.

    Let {X(t),tt0}, t00, be a TNH-RW process, having infinitesimal drift and infinitesimal variance

    A1(t)=β(t),A2(t)=σ2(t), (3.1)

    respectively, with β(t)R and σ(t)>0 continuous functions for all t. For the TNH-RW process, the results of Section 2 hold by choosing ζ1(x,t)=β(t) and ζ2(x,t)=σ2(t). We denote by rX(x,t|x0,t0) the transition pdf of X(t).

    The reflected Wiener process can be used as the diffusion approximation of the queueing system M/M/1 in heavy traffic conditions (see Giorno et al. [1], Kingman [24], Harrison [25]). It plays also an important role in economics and in finance (cf. Veestraeten [6], Linetsky [26]).

    Making use of the Fokker-Planck equation (2.1) with the boundary condition (2.2), for x00 one obtains the first two conditional moments of X(t):

    E[X(t)|X(t0)=x0]=x0+tt0β(u)du+12tt0σ2(u)rX(0,u|x0,t0)du,E[X2(t)|X(t0)=x0]=x20+tt0σ2(u)du+2tt0β(u)E[X(u)|X(t0)=x0]du.

    In the following proposition, we determine the transition pdf rX(x,t|x0,t0) in a special case.

    Proposition 3.1. Let X(t) be a TNH-RW process, having β(t)=γσ2(t), with γR, and σ(t)>0 in (3.1). One has:

    rX(x,t|x0,t0)=fX(x,t|x0,t0)x[e2γxFX(x,t|x0,t0)],x0,x00, (3.2)

    where

    fX(x,t|x0,t0)=12πVX(t|t0)exp{[xMX(t|x0,t0)]22VX(t|t0)},FX(x,t|x0,t0)=12[1+Erf(xMX(t|x0,t0)2VX(t|t0))], (3.3)

    with

    MX(t|x0,t0)=x0+tt0β(u)du,VX(t|t0)=tt0σ2(u)du (3.4)

    and Erf(x)=(2/π)x0ez2dz denoting the error function.

    Proof. Comparing (2.3) with (3.1), one has dh1(t)=β(t), h2(t)=0 and h1(t)h2(t)=σ2(t) for all t, from which

    h2(t)=c0,h1(t)=1ct0σ2(u)du,β(t)=dcσ2(t).

    Hence, for β(t)=γσ2(t), Eq. (3.2) follows from (2.4) by setting γ=d/c.

    Under the assumption of Proposition 3.1, making use of (3.3) in (3.2), we have:

    rX(x,t|x0,t0)=12πVX(t|t0)[exp{[xx0γVX(t|t0)]22VX(t|t0)}+e2γxexp{[x+x0+γVX(t|t0)]22VX(t|t0)}]γe2γxErfc(x+x0+γVX(t|t0)2VX(t|t0)),x0,x00, (3.5)

    where Erfc(x)=1Erf(x). We note that Eq. (3.5) for t0=0 is in agreement with Eq. (29) in Molini et al. [27].

    Corollary 3.1. Under the assumptions of Proposition 3.1, the following results hold:

    If β(t)=0, one has:

    E[X(t)|X(t0)=x0]=2VX(t|t0)πexp{x202VX(t|t0)}+x0Erf(x02VX(t|t0)),E[X2(t)|X(t0)=x0]=x20+VX(t|t0). (3.6)

    If β(t)=γσ2(t), with γ0, one obtains:

    E[X(t)|X(t0)=x0]=VX(t|t0)2πexp{[x0+γVX(t|t0)]22VX(t|t0)}+x0+γVX(t|t0)2[1+Erf(x0+γVX(t|t0)2VX(t|t0))]14γErfc(x0+γVX(t|t0)2VX(t|t0))+e2γx04γErfc(x0γVX(t|t0)2VX(t|t0)),E[X2(t)|X(t0)=x0]=[x0+γVX(t|t0)]2+VX(t|t0)2[1+Erf(x0+γVX(t|t0)2VX(t|t0))]+14γ2Erfc(x0+γVX(t|t0)2VX(t|t0))+VX(t|t0)2π(x0+γVX(t|t0)+1γ)×exp{[x0+γVX(t|t0)]22VX(t|t0)}e2γx02γ(x0γVX(t|t0)+12γ)Erfc(x0γVX(t|t0)2VX(t|t0)). (3.7)

    Proof. It follows making use of (3.5) in (2.5) with k=1,2, respectively.

    Corollary 3.2. The TNH-RW process, having A1(t)=γσ2(t) and A2(t)=σ2(t), with γR and σ(t)>0, admits the following asymptotic behaviors:

    1. when γ<0 and limt+σ2(t)=σ2 one has:

    WX(x)=limt+rX(x,t|x0,t0)=2|γ|e2|γ|x,x0. (3.8)

    2. when γ<0 and σ2(t) is a positive periodic function of period Q, one obtains:

    limn+rX(x,t+nQ|x0,t0)=2|γ|e2|γ|x,x0. (3.9)

    Proof. Eqs. (3.8) and (3.9) follow taking the appropriate limits in (3.5).

    Under the assumptions of Corollary 3.2, the TNH-RW process X(t) exhibits an exponential asymptotic behavior with mean and variance given by (2|γ|)1 and (2|γ|)2, respectively.

    For the TH-RW process X(t), in (3.1) we set β(t)=β and σ2(t)=σ2, with βR and σ>0. The scale function and the speed density, defined in (2.6), are:

    hX(x)=exp{2βσ2x},sX(x)=2σ2exp{2βσ2x}, (3.10)

    respectively. When β>0 (β0) the end-point + is an attracting (nonattracting) natural boundary. By choosing β(t)=β, σ2(t)=σ2 and γ=β/σ2, from Eq. (3.5) one has:

    rX(x,t|x0)=1σ2πt[exp{(xx0βt)22σ2t}+exp{2βxσ2}exp{(x+x0+βt)22σ2t}]βσ2exp{2βxσ2}Erfc(x+x0+βtσ2t),x0,x00, (3.11)

    and from (3.6) and (3.7) one obtains the first two conditional moments. Eq. (3.11) is in agreement with the expression given in Cox and Miller [28] obtained by using the method of images. We note that, if β<0, the TH-RW process X(t) admits an asymptotic behavior and the steady-state density is given in (3.8) with γ=β/σ2. Hence, for β<0 the steady-state density of the TH-RW process X(t) is exponential with mean E(X)=σ2/(2|β|) and variance Var(X)=σ4/(4β2).

    In Figure 1, we consider a TH-RW process X(t) in [0,+), having A1=β and A2=σ2, with β=0.6, σ=1, x0=5 and t0=0. Making use of Algorithm 4.2 in Buonocore et al. [29], we obtain a random sample of N=5104 observations of X(t). Then, we compare the histogram of the random sample with the transition pdf (3.11) as function of x (x0) for t=3 on the left and t=5 on the right. As shown in Figure 1, the histograms fit the exact transition densities (3.11) reasonably well.

    Figure 1.  For the TH-RW process X(t), with β=0.6, σ=1, x0=5, the transition pdf (3.11) is plotted (red curve) as function of x, for t=3 (on the left) and t=5 (on the right) and the histogram, obtained via the simulation of the sample paths, is superimposed over the density.

    For the TH-RW process, if 0S<x0 the FPT through S starting from x0 is not affected to reflecting boundary in the zero-state; in this case, if β<0 the ultimate FPT probability PX(S|x0)=1 and from (2.7) the FPT mean is t(X)1(S|x0)=(Sx0)/β. Moreover, if 0x0<S the probability PX(S|x0)=1 and making use of (2.8) one obtains:

    t(X)1(S|x0)={S2x20σ2,β=0,Sx0β+σ22β2[exp{2βSσ2}exp{2βx0σ2}],β0. (3.12)

    When 0x0<S, from (3.12) one has:

    t(X)1(S|x0) decreases as β increases and

    limβt(X)1(S|x0)=+,limβ+t(X)1(S|x0)=0;

    ● for β0, t(X)1(S|x0) decreases as σ2 increases; moreover, one has

    limσ20t(X)1(S|x0)={+,β0,(Sx0)/β,β>0,limσ2+t(X)1(S|x0)=0.

    In Figure 2, the FPT mean (3.12) of the TH-RW process is plotted for x0=5, S=10 and for different choices of β and σ2.

    Figure 2.  The FPT mean (3.12) of the TH-RW process is plotted for x0=5 and S=10 as function of β on the left and as function of σ2 on the right.

    Making use of (3.10) in Remark 2.1, one can derive some asymptotic behaviors for the FPT moments of the TH-RW process.

    Remark 3.1. For the TH-RW process X(t), when n=1,2, one has:

    1. limS+t(X)n(S|0)[t(X)1(S|0)]n=n!,β<0,

    2. limS0t(X)n(S|0)[t(X)1(S|0)]n=(1)nE2n(2n1)!!,

    where

    E0=1,E2n=n1j=0(2n2j)E2j,n=1,2

    denote the Euler numbers.

    Then, from Remark 3.1 it follows:

    ● for S+ one has:

    t(X)n(S|0)n![t(X)1(S|0)]n,β<0, (3.13)

    so that for β<0 the FPT density gX(S,t|0) of the TH-RW process exhibits an exponential asymptotic behavior for large boundaries;

    ● for S0 results:

    t(X)n(S|0)(1)nE2n(2n1)!![t(X)1(S|0)]n=m(X)n(S|0). (3.14)

    In particular, one has:

    m(X)2(S|0)=53[t(X)1(S|0)]2,m(X)3(S|0)=6115[t(X)1(S|0)]3.

    In Tables 35, t(X)1(S|0) is computed via (3.12), whereas t(X)2(S|0) and t(X)3(S|0) are numerically evaluated making use of the Siegert formula (2.8), with the scale function and the speed density given in (3.10). In particular, in Table 3, the mean t(X)1(S|0), the variance Var(X)(S|0), the coefficient of variation Cv(X)(S|0) and the skewness Σ(X)(S|0) of the FPT are listed for β=0.1, σ=1 and some choices of S>0. In agreement with the exponential asymptotic behavior, being β<0, the coefficient of variation and the skewness approach to 1 and to 2, respectively, as S increases.

    Table 3.  For the TH-RW process, with β=0.1 and σ=1, t(X)1(S|0), Var(X)(S|0), Cv(X)(S|0) and Σ(X)(S|0) are listed for increasing values of S>0.
    S t(X)1(S|0) Var(X)(S|0) Cv(X)(S|0) Σ(X)(S|0)
    10. 219.453 40104.8 0.912551 1.99238
    20. 2479.91 5.78195×106 0.96962 1.99921
    30. 19821.4 3.86673×108 0.992057 1.99995
    40. 148598 2.20066×1010 0.998305 2.0
    50. 1.10077×106 1.21093×1012 0.999681 2.00001
    60. 8.13709×106 6.62049×1013 0.999945 1.99999
    70. 6.01295×107 3.61549×1015 0.999991 2.00001
    80. 4.44305×108 1.97406×1017 0.999998 2.0
    90. 3.28300×109 1.07780×1019 0.999999 1.99995
    100. 2.42583×1010 5.88463×1020 1.0 1.99999

     | Show Table
    DownLoad: CSV
    Table 4.  For the TH-RW process, with β=0.1 and σ=1, t(X)1(S|0), t(X)2(S|0) and t(X)3(S|0) and their approximate values m(X)2(S|0) and m(X)3(S|0) are listed for decreasing values of S.
    S t(X)1(S|0) t(X)2(S|0) m(X)2(S|0) t(X)3(S|0) m(X)3(S|0)
    1.0 1.070138 1.929005 1.908659 5.105382 4.983768
    0.9 8.608682×101 1.247008 1.235157 2.651441 2.594469
    0.8 6.755435×101 7.670863×101 7.605985×101 1.278182 1.253714
    0.7 5.136899×101 4.430784×101 4.397956×101 5.606533×101 5.512413×101
    0.6 3.748426×101 2.356767×101 2.341783×101 2.173172×101 2.141832×101
    0.5 2.585459×101 1.120041×101 1.114100×101 7.114007×102 7.028321×102
    0.4 1.643534×101 4.521212×102 4.502006×102 1.823010×102 1.805405×102
    0.3 9.182733×102 1.409873×102 1.405376×102 3.171892×103 3.148868×103
    0.2 4.053871×102 2.744821×103 2.738978×103 2.722449×104 2.709245×104
    0.1 1.006700×102 1.690877×104 1.689075×104 4.159065×106 4.148957×106

     | Show Table
    DownLoad: CSV
    Table 5.  As in Table 4, for β=0.1 and σ=1.
    S t(X)1(S|0) t(X)2(S|0) m(X)2(S|0) t(X)3(S|0) m(X)3(S|0)
    1.0 9.365377×101 1.446255 1.461838 3.259422 3.340522
    0.9 7.635106×101 9.622582×101 9.715807×101 1.770458 1.810022
    0.8 6.071894×101 6.092237×101 6.144650×101 8.926616×101 9.103559×101
    0.7 4.679118×101 3.621786×101 3.649024×101 4.095229×101 4.166108×101
    0.6 3.460218×101 1.982750×101 1.995519×101 1.660226×101 1.684803×101
    0.5 2.418709×101 9.698262×102 9.750256×102 5.684286×102 5.754260×102
    0.4 1.558173×101 4.029243×102 4.046506×102 1.523487×102 1.538458×102
    0.3 8.822668×102 1.293173×102 1.297324×102 2.772401×103 2.792791×103
    0.2 3.947196×102 2.591186×103 2.596726×103 2.488771×104 2.500947×104
    0.1 9.933665×103 1.642874×104 1.644628×104 3.976568×106 3.986274×106

     | Show Table
    DownLoad: CSV

    Moreover, in Tables 4 and 5, for the TH-RW process with β=0.1,0.1 and σ=1, we compare the FPT moments t(X)2(S|0) and t(X)3(S|0) with the approximate values m(X)2(S|0) and m(X)3(S|0), respectively. We note that the goodness of the approximations improves as the boundary S approaches the reflecting zero-state.

    Let {Y(t),tt0}, t00, be a TNH-ROU process, having infinitesimal drift and infinitesimal variance

    B1(x,t)=α(t)x+β(t),B2(t)=σ2(t),xR, (4.1)

    with α(t)R, β(t)R and σ(t)>0 continuous functions for all t. For the TNH-ROU process, the results of Section 2 hold by choosing ζ1(x,t)=α(t)x+β(t) and ζ2(x,t)=σ2(t). Note that when α(t)=0 for all t, the process Y(t) identifies with the TNH-RW process X(t) with infinitesimal moments (3.1). We denote by rY(x,t|x0,t0) the transition pdf of Y(t).

    The reflected Ornstein-Uhlenbeck process arises as a diffusion approximation for population dynamics and for queueing systems (cf. Giorno et al. [1,30], Ward and Glynn [31]). Furthermore, the membrane potential evolution in neuronal diffusion models can be described by focusing the attention on the Ornstein-Uhlenbeck process confined by a lower reflecting boundary (cf., for instance, Buonocore et al. [9]). The reflected Ornstein-Uhlenbeck process can be also applied to the regulated financial market (see, Linetsky [26], Nie and Linetsky [32]).

    Recalling the Fokker-Planck equation (2.1), with the boundary condition (2.2), for x00 one has the first two conditional moments of Y(t):

    E[Y(t)|Y(t0)=x0]=x0eA(t|t0)+tt0eA(t|u)β(u)du+12tt0eA(t|u)σ2(u)rY(0,u|x0,t0)du,E[Y2(t)|Y(t0)=x0]=x20e2A(t|t0)+tt0σ2(u)e2A(t|u)du+2tt0β(u)e2A(t|u)E[Y(u)|Y(t0)=x0]du,

    being

    A(t|t0)=tt0α(z)dz. (4.2)

    We now determine the transition pdf rY(x,t|x0,t0) in a special case.

    Proposition 4.1. Let Y(t) be a TNH-ROU process, having β(t)=γσ2(t)eA(t|0), with γR, σ(t)>0 and A(t|t0) given in (4.2). One has:

    rY(x,t|x0,t0)=fY(x,t|x0,t0)x[exp{2γxeA(t|0)}FY(x,t|x0,t0)],x0,x00, (4.3)

    where

    fY(x,t|x0,t0)=12πVY(t|t0)exp{[xMY(t|x0,t0)]22VY(t|t0)},FY(x,t|x0,t0)=12[1+Erf(xMY(t|x0,t0)2VY(t|t0))] (4.4)

    with

    MY(t|x0,t0)=x0eA(t|t0)+tt0β(u)eA(t|u)du,VY(t|t0)=tt0σ2(u)e2A(t|u)du. (4.5)

    Proof. Comparing (2.3) with (4.1), one has

    h2(t)h2(t)=α(t),h1(t)h2(t)h1(t)h2(t)=σ2(t),dσ2(t)h2(t)=β(t)

    for all t0, from which

    h2(t)=ceA(t|0),h1(t)=eA(t|0)ct0σ2(u)e2A(u|0)du,β(t)=dσ2(t)ceA(t|0),

    with c0. Then, if β(t)=γσ2(t)eA(t|0), Eq. (4.3) follows from (2.4) by setting γ=d/c.

    Under the assumptions of Proposition 4.1, making use of (4.4) in (4.3), for x0,x00 we have:

    rY(x,t|x0,t0)=12πVY(t|t0)[exp{[xH(t|x0,t0)]22VY(t|t0)}+exp{2γxeA(t|0)}exp{[x+H(t|x0,t0)]22VY(t|t0)}]γeA(t|0)exp{2γxeA(t|0)}Erfc(x+H(t|x0,t0)2VY(t|t0)), (4.6)

    where we have set:

    H(t|x0,t0)=x0eA(t|t0)+γeA(t|0)VY(t|t0). (4.7)

    Corollary 4.1. Under the assumptions of Proposition 4.1, the following results hold:

    when β(t)=0, one obtains

    E[Y(t)|Y(t0)=x0]=2VY(t|t0)πexp{x20e2A(t|t0)2VY(t|t0)}+x0eA(t|t0)Erf(x0eA(t|t0)2VY(t|t0)),E[Y2(t)|Y(t0)=x0]=x20e2A(t|t0)+VY(t|t0). (4.8)

    when β(t)=γσ2(t)eA(t|0), with γ0, one has

    E[Y(t)|Y(t0)=x0]=VY(t|t0)2πexp{H2(t|x0,t0)2VY(t|t0)}+H(t|x0,t0)2[1+Erf(H(t|x0,t0)2VY(t|t0))]eA(t|0)4γErfc(H(t|x0,t0)2VY(t|t0))+eA(t|0)4γexp{2γx0eA(t0|0)}Erfc(H(t|x0,t0)2γeA(t|0)VY(t|t0)2VY(t|t0)),E[Y2(t)|Y(t0)=x0]=H2(t|x0,t0)+VY(t|t0)2[1+Erf(H(t|x0,t0)2VY(t|t0))]+e2A(t|0)4γ2Erfc(H(t|x0,t0)2VY(t|t0))+VY(t|t0)2π(H(t|x0,t0)+eA(t|0)γ)exp{H2(t|x0,t0)2VY(t|t0)}e2A(t|0)4γ2exp{2γx0eA(t0|0)}×{14γ2e2A(t|0)VY(t|t0)+2γeA(t|0)H(t|x0,t0)}Erfc(H(t|x0,t0)2γeA(t|0)VY(t|t0)2VY(t|t0)), (4.9)

    with H(t|x0,t0) given in (4.7).

    Proof. It follows making use of (4.6) in (2.5) for k=1,2, respectively.

    We note that if α(t)=0 for all t, the conditional moments (4.8) and (4.9) identify with conditional moments (3.6) and (3.7) of the TNH-RW process with β(t)=γσ2(t), being H(t|x0,t0)=x0+γVX(t|t0) in (4.7).

    Corollary 4.2. For the TNH-ROU process, having B1(x)=αx and B2(t)=σ2(t), with αR and σ(t)>0, the following asymptotic behaviors hold:

    1. when α<0 and limt+σ2(t)=σ2 one has:

    WY(x)=limt+rY(x,t|x0,t0)=2|α|πσ2exp{|α|x2σ2},x0 (4.10)

    and the first two asymptotic moments are

    E(Y)=limt+E[Y(t)|Y(t0)=x0]=σπ|α|,E(Y2)=limt+E[Y2(t)|Y(t0)=x0]=σ22|α|

    2. when α<0 and σ2(t) is a positive periodic function of period Q, one obtains:

    limn+rY(x,t+nQ|x0,t0)=2πω1(t)exp{x22ω1(t)},x0, (4.11)

    with

    ω1(t)=VY(t+Q|t)1e2|α|Q,t0,

    and the first two asymptotic moments are

    limn+E[Y(t+nQ)|Y(t0)=x0]=2ω1(t)π,limn+E[Y2(t+nQ)|Y(t0)=x0]=ω1(t).

    Proof. Eqs. (4.10) and (4.11) follow from (4.6) by setting α(t)=α and γ=0. When α<0, in the case 1. one has limt+VY(t|t0)=σ2/(2|α|), whereas in the case 2. it results limn+VY(t+nQ|t0)=ω1(t).

    For the TH-ROU process Y(t), in (4.1) we set α(t)=α, β(t)=β, σ2(t)=σ2, with α0, βR and σ>0. The scale function and the speed density, defined in (2.6), are:

    hY(x)=exp{ασ2(x2+2βαx)},sY(x)=2σ2exp{ασ2(x2+2βαx)}, (4.12)

    respectively. When α>0 (α<0) the end-point + is an attracting (nonattracting) natural boundary.

    Proposition 4.2. Let r(Y)λ(x|x0)=+0eλtrY(x,t|x0)dt (Reλ>0) be the LT of the transition pdf of the TH-ROU process Y(t). One has

    r(Y)λ(x|x0)={2λ|α|1σπ|α|Γ(λ2|α|)Γ(12+λ2|α|)exp{|α|2σ2[(x+βα)2(x0+βα)2]}×Dλ|α|(2|α|σ[x0x+βα])[Dλ|α|(2|α|σ[x0x+βα])+Dλ|α|1(2|α|σβα)Dλ|α|1(2|α|σβα)Dλ|α|(2|α|σ[x0x+βα])],α<0,2λασπαΓ(1+λ2α)Γ(12+λ2α)exp{α2σ2[(x0+βα)2(x+βα)2]}×Dλα1(2ασ[x0x+βα])[Dλα1(2ασ[x0x+βα])+Dλα(2ασβα)Dλα(2ασβα)Dλα1(2ασ[x0x+βα])],α>0, (4.13)

    where x0x=max(x0,x), x0x=min(x0,x), Dν(z) is the parabolic cylinder function defined as

    Dν(z)=2ν/2ez2/4{πΓ(1ν2)Φ(ν2,12;z22)z2πΓ(ν2)Φ(1ν2,32;z22)}

    and

    Φ(a,c;x)=1++n=1(a)n(c)nxnn!,

    is the Kummer's confluent hypergeometric function, with (a)0=1 and (a)n=a(a+1)(a+n1) for n=1,2,

    Proof. The proof is given in A.

    Eq. (4.13) can be used to analyze the asymptotic behavior of the TH-ROU process.

    Corollary 4.3. For α<0, the TH-ROU process Y(t) admits an asymptotic behavior. The steady-state density is

    WY(x)=limt+rY(x,t|x0)=2σ|α|πexp{|α|σ2(x+βα)2}1Erfc(|α|σβα),x0, (4.14)

    and the asymptotic moments of the first and second order are:

    E(Y)=β|α|+σπ|α|exp{|α|σ2(βα)2}1Erfc(|α|σβα),E(Y2)=(βα)2+σ22|α|+β|α|σπ|α|exp{|α|σ2(βα)2}1Erfc(|α|σβα) (4.15)

    Proof. If α<0, from (4.13) one has:

    WY(x)=limλ0λr(Y)λ(x|x0)=1σ|α|πexp{|α|2σ2[(x+βα)2(x0+βα)2]}D0(2|α|σ[x0x+βα])×[D0(2|α|σ[x0x+βα])+D1(2|α|σβα)D1(2|α|σβα)D0(2|α|σ[x0x+βα])],x0. (4.16)

    Since (cf. Gradshteyn and Ryzhik [33], p. 1030, no. 9.251 and no. 9.254)

    D0(x)=ex2/4,D1(x)=π2ex2/4Erfc(x2), (4.17)

    from (4.16) one obtains (4.14). The asymptotic moments (4.15) follow from (4.14).

    Eq. (4.14) is in agreement with the result of the Proposition 1 in Ward and Glinn [31]; moreover, for β=0 Eq. (4.14) identifies with (4.10).

    The inverse LT of (4.13) is obtainable only for β=0; in this case, the following result holds.

    Proposition 4.3. For the TH-ROU process Y(t), having B1(x)=αx and B2=σ2, with α0 and σ>0, one has:

    rY(x,t|x0)=απσ2(e2αt1)[exp{α(xx0eαt)2σ2(e2αt1)}+exp{α(x+x0eαt)2σ2(e2αt1)}],x0,x00. (4.18)

    Proof. It follows from (4.6) with γ=0, by setting t0=0, A(t|0)=eαt and VY(t|0)=σ2(e2αt1)/α. Alternatively, Eq. (4.18) can be obtained from (4.13) and (A.4) with β=0, by noting that r(Y)λ(x|x0)=f(Y)λ(x|x0)+f(Y)λ(x|x0).

    In Figure 3, we consider a TH-ROU process Y(t) in [0,+), having B1(x)=αx+β and B2=σ2, with α=0.5, β=0, σ=1, x0=5 and t0=0. Using the Algorithm 4.1 in Buonocore et al. [29], we obtain a random sample of N=5104 observations of Y(t). Then, we compare the histogram of the random sample with the transition pdf (4.18) as function of x (x0) for t=2 on the left and t=4 on the right. Figure 3 shows the good agreement between the histograms obtained via simulation and the exact density (4.18).

    Figure 3.  For the TH-ROU process Y(t), with α=0.5, β=0, σ=1, x0=5, the transition pdf (4.18) is plotted (red curve) as function of x, for t=2 (on the left) and t=4 (on the right) and the histogram, obtained via the simulation of the sample paths, is superimposed over the density.

    Under the assumption of Proposition 4.3, the first two conditional moments of the TH-ROU process can be obtained from (4.8).

    For the TH-ROU process, if 0S<x0 the FPT through S starting from x0 is not affected to reflecting boundary in the zero-state; in this case, for α<0 the ultimate FPT probability PY(S|x0)=1 and from (2.7) the FPT mean is

    t(Y)1(S|x0)=1|α|{ψ1(|α|σ(S+βα))ψ1(|α|σ(x0+βα))+π2[Erfi(|α|σ(x0+βα))Erfi(|α|σ(S+βα)]},0S<x0, (4.19)

    where

    ψ1(x)=+k=02kx2k+2(k+1)(2k+1)!!,Erfi(x)=2πx0ez2dz=2π+k=0x2k+1(2k+1)k!

    Moreover, if 0x0<S the probability PY(S|x0)=1 and making use of (2.8) one obtains:

    t(Y)1(S|x0)={1|α|{ψ1(|α|σ(S+βα))ψ1(|α|σ(x0+βα))π2Erf(βσ|α|)×[Erfi(|α|σ(S+βα))Erfi(|α|σ(x0+βα)]},α<0,1α{ψ2(ασ(S+βα))ψ2(ασ(x0+βα))π2Erfi(βσα)×[Erf(ασ(S+βα))Erf(ασ(x0+βα)]},α>0, (4.20)

    where

    ψ2(z)=+k=0(1)k2kz2k+2(k+1)(2k+1)!!

    When 0x0<S, from (4.20) one has:

    t(Y)1(S|x0) decreases as β increases and

    limβt(Y)1(S|x0)=+,limβ+t(X)1(S|x0)=0;

    ● for α<0, t(Y)1(S|x0) decreases as σ2 increases; moreover, one has:

    limσ20t(Y)1(S|x0)={+,[α<0,βR]or[α>0,β<0],1αln(αS+βαx0+β),[α>0,β0],limσ2+t(Y)1(S|x0)=0.

    In Figures 4 and 5, the FPT mean (4.20) of the TH-ROU process is plotted for x0=5, S=10 and for different choices of β and σ2, with α=0.02,0.02, respectively.

    Figure 4.  The FPT mean (4.20) is plotted for α=0.02, x0=5 and S=10 as function of β on the left and as function of σ2 on the right.
    Figure 5.  As in Figure 4 with α=0.02.

    Making use of (4.12) in Remark 2.1, one can derive some asymptotic behaviors for the FPT moments of the TH-ROU process.

    Remark 4.1. For the TH-ROU process Y(t), when n=1,2, one has:

    1. limS+t(Y)n(S|0)[t(Y)1(S|0)]n=n!,α<0,

    2. limS0t(Y)n(S|0)[t(Y)1(S|0)]n=(1)nE2n(2n1)!!,

    where E0,E1, are the Euler numbers.

    Then, from Remark 4.1, it follows:

    ● for S+ one has:

    t(Y)n(S|0)n![t(Y)1(S|0)]n,α<0, (4.21)

    so that for α<0 the FPT density gY(S,t|0) of the TH-ROU process exhibits an exponential trend for large boundary;

    ● for S0 it results:

    t(Y)n(S|0)(1)nE2n(2n1)!![t(Y)1(S|0)]n=m(Y)n(S|0). (4.22)

    In particular, one has:

    m(Y)2(S|0)=53[t(Y)1(S|0)]2,m(Y)3(S|0)=6115[t(Y)1(S|0)]3.

    In Tables 69, t(Y)1(S|0) is computed via (4.20), whereas t(Y)2(S|0) and t(Y)3(S|0) are numerically evaluated making use of (2.8) and (4.12). In particular, in Tables 6 and 7, the mean t(Y)1(S|0), the variance Var(Y)(S|0), the coefficient of variation Cv(Y)(S|0) and the skewness Σ(Y)(S|0) of the FPT are listed for α=0.02, β=0.1,0.1, σ=1 and some choices of S>0. Being α<0, the coefficient of variation and the skewness approach to 1 and to 2, respectively, as S increases.

    Table 6.  For the TH-ROU process, with α=0.02, β=0.1 and σ=1, t(Y)1(S|0), Var(Y)(S|0), Cv(Y)(S|0) and Σ(Y)(S|0) are listed for increasing values of S>0.
    S t(Y)1(S|0) Var(Y)(S|0) Cv(Y)(S|0) Σ(Y)(S|0)
    10. 635.964 380988. 0.970562 1.99926
    20. 1.11639×106 1.24622×1012 0.999956 1.99998
    30. 1.26765×1011 1.60694×1022 1.0 1.99997
    40. 8.68434×1017 7.54177×1035 1.0 1.99991
    50. 3.43245×1026 1.17817×1053 1.0 1.99998
    60. 7.67474×1036 5.89016×1073 1.0 1.99996
    70. 9.60516×1048 9.22590×1097 1.0 1.99992
    80. 6.68550×1062 4.46959×10125 1.0 1.99984
    90. 2.57707×1078 6.64129×10156 1.0 2.00008
    100. 5.48553×1095 3.00911×10191 1.0 1.99998

     | Show Table
    DownLoad: CSV
    Table 7.  As in Table 6 for α=0.02, β=0.1 and σ=1.
    S t(Y)1(S|0) Var(Y)(S|0) Cv(Y)(S|0) Σ(Y)(S|0)
    10. 102.243 7056.41 0.821592 1.96257
    20. 3664.17 1.29250×107 0.981159 1.99973
    30. 5.92059×106 3.50521×1013 0.999982 1.99990
    40. 6.72233×1011 4.51897×1023 1.0 1.99990
    50. 4.60528×1018 2.12086×1037 1.0 1.99993
    60. 1.82022×1027 3.31320×1054 1.0 2.00001
    70. 4.06990×1037 1.65640×1075 1.0 2.00009
    80. 5.09359×1049 2.59446×1099 1.0 1.99994
    90. 3.54531×1063 1.25692×10127 1.0 1.99995
    100. 1.36661×1079 1.86763×10158 1.0 1.99993

     | Show Table
    DownLoad: CSV
    Table 8.  For the TH-ROU with α=0.02, β=0.1 and σ=1, t(Y)1(S|0), t(Y)2(S|0) and t(Y)3(S|0) and their approximate values m(Y)2(S|0) and m(Y)3(S|0) are listed for decreasing values of S.
    S t(Y)1(S|0) t(Y)2(S|0) m(Y)2(S|0) t(Y)3(S|0) m(Y)3(S|0)
    1.0 1.077553 1.957892 1.935202 5.224688 5.088091
    0.9 8.656790×101 1.262062 1.249 2.701365 2.63821
    0.8 6.785138×101 7.743699×101 7.673017×101 1.297103 1.270324
    0.7 5.154121×101 4.462854×101 4.427493×101 5.669774×101 5.568040×101
    0.6 3.757621×101 2.369247×101 2.353286×101 2.191101×101 2.157633×101
    0.5 2.589846×101 1.124143×101 1.117884×101 7.154595×102 7.064161×102
    0.4 1.645312×101 4.531770×102 4.511752×102 1.829640×102 1.811271×102
    0.3 9.188300×102 1.411719×102 1.407081×102 3.178358×103 3.154599×103
    0.2 4.054959×102 2.746412×103 2.740449×103 2.724907×104 2.711428×104
    0.1 1.006767×102 1.691121×104 1.689301×104 4.160001×106 4.149790×106

     | Show Table
    DownLoad: CSV
    Table 9.  As in Table 8 for α=0.02, β=0.1 and σ=1.
    S t(Y)1(S|0) t(Y)2(S|0) m(Y)2(S|0) t(Y)3(S|0) m(Y)3(S|0)
    1.0 9.426077×101 1.466641 1.480849 3.331454 3.405897
    0.9 7.675284×101 9.732598×101 9.818331×101 1.802163 1.838747
    0.8 6.097203×101 6.147356×101 6.195981×101 9.053004×101 9.217869×101
    0.7 4.694088×101 3.646917×101 3.672411×101 4.139661×101 4.206223×101
    0.6 3.468374×101 1.992877×101 2.004936×101 1.673475×101 1.696744×101
    0.5 2.422679×101 9.732731×102 9.782287×102 5.715830×102 5.782639×102
    0.4 1.559815×101 4.038428×102 4.055036×102 1.528907×102 1.543325×102
    0.3 8.827911×102 1.294836×102 1.298867×102 2.777959×103 2.797773×103
    0.2 3.948242×102 2.592670×103 2.598102×103 2.490994×104 2.502936×104
    0.1 9.934325×103 1.643110×104 1.644847×104 3.977457×106 3.987069×106

     | Show Table
    DownLoad: CSV

    Moreover, in Tables 8 and 9, for the TH-ROU process with α=0.02, β=0.1,0.1 and σ=1 we compare the FPT moments t(Y)2(S|0) and t(Y)3(S|0) with the approximate values m(Y)2(S|0) and m(Y)3(S|0), respectively. From Tables 8 and 9, we note that the goodness of the approximations improves as the boundary S approaches the reflecting zero-state.

    Let {Z(t),tt0}, t00, be a TNH-RF process, having the following infinitesimal drift and infinitesimal variance

    C1(x,t)=α(t)x+β(t),C2(x,t)=2r(t)x, (5.1)

    with α(t)R, β(t)>0 and r(t)>0 continuous functions for all t, defined in the state-space [0,+) with a zero-flux condition in the zero-state. For the TNH-RF process, the results of Section 2 hold by choosing ζ1(x,t)=α(t)x+β(t) and ζ2(x,t)=2r(t)x. We denote with rZ(x,t|x0,t0) the transition pdf of Z(t) .

    Feller diffusion process is applied in population dynamics to model the growth of a population (cf. Ricciardi et al. [15], Giorno and Nobile [34]). This process is also used in queueing systems to describe the number of customers in a queue (cf. Di Crescenzo and Nobile [35]), in neurobiology to analyze the input-output behavior of single neurons (see, for instance, Giorno et al. [36], Ditlevsen and Lánský [37]), in mathematical finance to model asset prices, market indices, interest rates and stochastic volatility (see, Tian and Zhang [38], Cox et al. [39], Di Nardo and D'Onofrio [40]). We emphasize that in the mathematical finance, the Feller process is also known as Cox-Ingersoll-Ross (CIR) model.

    Making use of Eq. (2.1), with the zero-flux condition in the zero-state (2.2), for x_0\geq 0 one obtains the conditional mean and the conditional variance of Z(t) :

    \begin{equation} \begin{split} &{\rm E}[Z(t)|Z(t_0) = x_0] = x_0\,e^{A(t|t_0)}+\int_{t_0}^t\beta(u)\,e^{A(t|u)}\;du,\\ &{\rm Var}[ Z(t)| Z(t_0) = x_0] = 2\,x_0\,e^{2\,A(t|t_0)}\,R(t|t_0) +2\,e^{A(t|t_0)}\,\int_{t_0}^t\beta(u)\,e^{A(t|u)}\bigl[R(t|t_0)-R(u|t_0)\bigr]\;du, \end{split} \end{equation} (5.2)

    with A(t|t_0) defined in (4.2) and

    \begin{equation} R(t|t_0) = \int_{t_0}^t r(\tau)\,e^{-A(\tau|t_0)}\,d\tau,\qquad t\geq t_0. \end{equation} (5.3)

    We note that when \alpha(t)\neq 0 , the conditional mean M_Y(t|x_0, t_0) of the unrestricted Ornstein-Uhlenbeck process, given in (4.5), identifies with the conditional average of the Feller process Z(t) , given in (5.2). Similarly, when \alpha(t) = 0 for all t , the conditional average of the Feller process is equal to the conditional mean M_X(t|x_0, t_0) of the unrestricted Wiener process, given in (3.4).

    In the following proposition, we consider the transition pdf r_Z(x, t|x_0, t_0) in a special case.

    Proposition 5.1. Let Z(t) be a TNH-RF process, with \alpha(t)\in\mathbb{R} , r(t) > 0 and \beta(t) = \xi\, r(t) , with \xi > 0 , in (5.1). The following results hold:

    \begin{equation} r_Z(x,t|x_0,t_0) = \left\{\begin{array}{ll} {{1\over x\,\Gamma(\xi)}}\Bigl[ {x\,e^{-A(t|t_0)}\over R(t|t_0)}\Bigr]^{\xi}\exp\Bigl\{-{x\,e^{-A(t|t_0)}\over R(t|t_0)}\Bigr\} ,&x_0 = 0,\\ \\ {e^{-A(t|t_0)}\over R(t|t_0)}\,\Bigl[ {x\, e^{-A(t|t_0)}\over x_0}\Bigr]^{(\xi-1)/2}\,\exp\Bigl\{-{x_0+x\,e^{-A(t|t_0)}\over R(t|t_0)}\Bigr\} \,I_{\xi-1}\biggl[ {2\sqrt{x\,x_0\,e^{-A(t|t_0)\,}}\over R(t|t_0)}\biggr], &x_0 > 0, \end{array}\right. \end{equation} (5.4)

    where

    I_{\nu}(z) = \sum\limits_{k = 0}^{+\infty}{1\over k!\,\Gamma(\nu+k+1)}\,\Bigl({z\over 2}\Bigr)^{2k+\nu}, \label{Bessel_I}

    is the modified Bessel function of the first kind, with \Gamma(\nu) denoting the Euler gamma function. Moreover, one has

    \begin{equation} \begin{split} &E[Z(t)|Z(t_0) = x_0] = e^{A(t|t_0)}\,\Bigl[x_0+\xi\,R(t|t_0)\Bigr],\\ &E[Z^2(t)|Z(t_0) = x_0] = e^{2A(t|t_0)}\,\Bigl[x_0^2+2(\xi+1)R(t|t_0)x_0+\xi(\xi+1)R^2(t|t_0)\Bigr], \end{split} \end{equation} (5.5)

    where A(t|t_0) and R(t|t_0) are defined in (4.2) and (5.3), respectively.

    Proof. Eq. (5.4) follows as in Giorno and Nobile [34] and Masoliver [41]. Moreover, relations (5.5) are obtainable from (5.2).

    Since, for fixed \nu , when z\to 0 one has

    \begin{equation} I_{\nu}(z)\sim {1\over \Gamma(\nu+1)}\,\Bigl({z\over 2}\Bigr)^{\nu}, \qquad \nu\neq -1,-2,\ldots, \end{equation} (5.6)

    the first formula in (5.4) follows from the second expression as x_0\downarrow 0 .

    Corollary 5.1. The TNH-RF process Z(t) , having C_1(x, t) = \alpha\, x+\xi\, r(t) and C_2(x, t) = 2\, r(t)\, x , with \alpha\in\mathbb{R} , \xi > 0 and r(t) > 0 , admits the following asymptotic behaviors:

    1. when \alpha < 0 and \lim_{t\to +\infty}r(t) = r , with r > 0 , one has:

    \begin{equation} W_Z(x) = \lim\limits_{t\to+\infty}r_Z(x,t|x_0,t_0) = {1\over x\,\Gamma(\xi)}\Bigl({|\alpha|\,x\over r}\Bigr)^{\xi}\,\exp\Bigl\{-{|\alpha|\,x\over r}\Bigr\},\quad x > 0, \end{equation} (5.7)

    and the asymptotic moments are

    E(Z^k) = \lim\limits_{t\to +\infty} E[Z^k(t)|Z(t_0) = x_0] = \Bigl({r\over |\alpha|}\Bigr)^k\,{\Gamma(k+\xi)\over\Gamma(\xi)},\quad k = 1,2,\ldots

    2. when \alpha < 0 and r(t) is a positive periodic function of period Q , one obtains:

    \begin{equation} \lim\limits_{n\to+\infty}r_Z(x,t+nQ|x_0,t_0) = {1\over x\,\Gamma(\xi)}\, \Bigl[\omega_2(t)\,x\Bigr]^{\xi}\,\exp\Bigl\{-\omega_2(t)\,x\Bigr\},\quad x > 0, \end{equation} (5.8)

    with

    \omega_2(t) = {e^{|\alpha|\,Q}-1\over R(t+Q|t)},\qquad t\geq 0,

    and the asymptotic moments are

    \lim\limits_{n\to +\infty} E[Z^k(t+nQ)|Z(t_0) = x_0] = \bigl[\omega_2(t)\bigr]^{-k}\,{\Gamma(k+\xi)\over\Gamma(\xi)},\quad k = 1,2,\ldots

    Proof. Eq. (5.7) follows from (5.4) making use of (5.6) and by noting that

    \lim\limits_{t\to +\infty}R(t|t_0) = +\infty,\quad \lim\limits_{t\to +\infty}[e^{A(t|t_0)}R(t|t_0)] = {r\over |\alpha|}, \quad \lim\limits_{t\to +\infty}{e^{-A(t|t_0)/2}\over R(t|t_0)} = 0.

    Similarly, since \lim_{n\to+\infty}\Bigl[e^{-A(t+nQ|t_0)}/R(t+nQ|t_0)\Bigr] = \omega_2(t) , one can be obtain (5.8).

    We note that (5.7) is a gamma density of parameters \xi and r/|\alpha| , that is a decreasing function of x when 0 < \xi\leq 1 , whereas it has a single maximum in x = r(\xi-1)/|\alpha| for \xi > 1 . Similarly, (5.8) is a non-homogeneous gamma density.

    In Propositions 5.2 and 5.3, we prove some relations between the transition pdf of TNH-RF process with \beta(t) = r(t)/2 and \beta(t) = 3\, r(t)/2 in (5.1) and the transition pdf of Wiener and of Ornstein-Uhlenbeck processes under suitable conditions on the zero-state and for specific choices of the infinitesimal moments.

    Proposition 5.2. Let Z(t) be a TNH-RF process with C_1(x, t) = \alpha(t)\, x+r(t)/2 and C_2(x, t) = 2\, r(t)\, x , where \alpha(t)\in\mathbb{R} and r(t) > 0 .

    1. If \alpha(t) = 0 for all t , one has

    \begin{equation} r_Z(x,t|x_0,t_0) = {1\over 2\sqrt{x}}\,r_X(\sqrt{x},t|\sqrt{x_0},t_0),\qquad x_0\geq 0,x > 0, \end{equation} (5.9)

    where r_X(x, t|x_0, t_0) denotes the transition pdf of the TNH-RW process with A_1 = 0 and A_2(t) = r(t)/2 .

    2. If \alpha(t) is not always zero, it follows:

    \begin{equation} r_Z(x,t|x_0,t_0) = {1\over 2\sqrt{x}}\,r_Y(\sqrt{x},t|\sqrt{x_0},t_0),\qquad x_0\geq 0,x > 0, \end{equation} (5.10)

    where r_Y(x, t|x_0, t_0) denotes the transition pdf of the TNH-ROU process with B_1(x, t) = \alpha(t)\, x/2 and B_2(t) = r(t)/2 .

    Proof. For the TNH-RF process Z(t) , by setting \beta(t) = r(t)/2 in (5.4) and recalling that

    I_{-1/2}(x) = \sqrt{2\over\pi}\,{\cosh(x)\over\sqrt{x}},

    for x_0\geq 0, x > 0 one has:

    \begin{eqnarray} &&r_Z(x,t|x_0,t_0) = \left\{\begin{array}{ll} {1\over 2\sqrt{\pi\,x\,\widetilde R(t|t_0) }}\biggl[\exp\biggl\{-{\bigl(\sqrt{x}-\sqrt{x_0}\bigr)^2\over \widetilde R(t|t_0)}\biggr\} +\exp\biggl\{-{\bigl(\sqrt{x}+\sqrt{x_0}\bigr)^2\over \widetilde R(t|t_0)}\biggr\}\biggr], &\alpha(t) = 0,\\ \\ {e^{-A(t|t_0)/2}\over 2\sqrt{\pi\,x\,R(t|t_0)}}\,\biggl[\exp\biggl\{-{\bigl(\sqrt{x\,e^{-A(t|t_0)}}-\sqrt{x_0}\bigr)^2\over R(t|t_0)}\biggr\} +\exp\biggl\{-{\bigl(\sqrt{x\,e^{-A(t|t_0)}}+\sqrt{x_0}\bigr)^2\over R(t|t_0)}\biggr\} \biggr], &\alpha(t)\neq 0, \end{array}\right. \end{eqnarray} (5.11)

    with \widetilde R(t|t_0) = \int_{t_0}^tr(\theta)\, d\theta and with A(t|t_0) and R(t|t_0) given in (4.2) and (5.3), respectively. We now analyze separately the cases 1. and 2.

    1. For the TNH-RW process X(t) , defined in (3.1), with \beta(t) = 0 and \sigma^2(t) = r(t)/2 , one has V_X(t|t_0) = \widetilde R(t|t_0)/2 , so that from (3.5) with \gamma = 0 one obtains:

    \begin{equation} r_X(\sqrt{x},t|\sqrt{x_0},t_0) = {1\over\sqrt{\pi\widetilde R(t|t_0) }}\biggl[\exp\biggl\{-{\bigl(\sqrt{x}-\sqrt{x_0}\bigr)^2\over \widetilde R(t|t_0)}\biggr\} +\exp\biggl\{-{\bigl(\sqrt{x}+\sqrt{x_0}\bigr)^2\over \widetilde R(t|t_0)}\biggr\}\biggr] \end{equation} (5.12)

    for x_0\geq 0 and x\geq 0 . Then, by comparing the first of (5.11) with (5.12), for x_0\geq 0 and x > 0 relation (5.9) follows.

    2. In the TNH-ROU process Y(t) , defined in (4.1), we set \beta(t) = 0 , \sigma^2(t) = r(t)/2 and we change \alpha(t) into \alpha(t)/2 ; by virtue of (4.5) and (5.3), one has V_Y(t|t_0) = R(t|t_0)e^{A(t|t_0)}/2 , so that from (4.6) with \gamma = 0 one obtains:

    \begin{eqnarray} &&r_Y({\sqrt x},t|\sqrt{x_0},t_0) = {e^{-A(t|t_0)/2}\over \sqrt{\pi\,R(t|t_0)}}\,\biggl[\exp\biggl\{-{\bigl(\sqrt{x\,e^{-A(t|t_0)}}-\sqrt{x_0}\bigr)^2\over R(t|t_0)}\biggr\} +\exp\biggl\{-{\bigl(\sqrt{x\,e^{-A(t|t_0)}}+\sqrt{x_0}\bigr)^2\over R(t|t_0)}\biggr\} \biggr],\\ && x_0\geq 0,x\geq 0. \end{eqnarray} (5.13)

    Hence, for x_0\geq 0 and x > 0 , Eq. (5.10) follows by comparing the second of (5.11) with (5.13).

    Proposition 5.3. Let Z(t) be a TNH-RF process with C_1(t) = \alpha(t)\, x+3r(t)/2 and C_2(x, t) = 2\, r(t)\, x , where \alpha(t)\in\mathbb{R} and r(t) > 0 .

    1. If \alpha(t) = 0 for all t , one has

    \begin{equation} r_Z(x,t|x_0,t_0) = {1\over 2\sqrt{x_0}}\,a_X(\sqrt{x},t|\sqrt{x_0},t_0),\qquad x_0 > 0,x > 0, \end{equation} (5.14)

    where a_X(x, t|x_0, t_0) denotes the transition pdf of the inhomogeneous Wiener process with A_1 = 0 and A_2(t) = r(t)/2 , restricted to (0, +\infty) with an absorbing boundary in the zero-state.

    2. If \alpha(t) is not always zero, it follows:

    \begin{equation} r_Z(x,t|x_0,t_0) = {e^{-A(t|t_0)/2}\over 2\sqrt{x_0}}\,a_Y(\sqrt{x},t|\sqrt{x_0},t_0),\qquad x_0 > 0,x > 0, \end{equation} (5.15)

    where a_Y(x, t|x_0, t_0) denotes the transition pdf of the inhomogeneous Ornstein-Uhlenbeck process with B_1(x, t) = \alpha(t)\, x/2 and B_2(t) = r(t)/2 , restricted to (0, +\infty) with an absorbing boundary in the zero-state.

    Proof. For the TNH-RF process Z(t) , by setting \beta(t) = 3r(t)/2 in (5.4) and recalling that

    I_{1/2}(x) = \sqrt{2\over\pi}\,{\sinh(x)\over\sqrt{x}},

    for x_0 > 0, x > 0 one has:

    \begin{eqnarray} &&r_Z(x,t|x_0,t_0) = \left\{\begin{array}{ll} {1\over 2\sqrt{\pi\,x_0\,\widetilde R(t|t_0) }}\biggl[\exp\biggl\{-{\bigl(\sqrt{x}-\sqrt{x_0}\bigr)^2\over \widetilde R(t|t_0)}\biggr\} -\exp\biggl\{-{\bigl(\sqrt{x}+\sqrt{x_0}\bigr)^2\over \widetilde R(t|t_0)}\biggr\}\biggr], &\alpha(t) = 0,\\ \\ {e^{-A(t|t_0)}\over 2\sqrt{\pi\,x_0\,R(t|t_0)}}\,\biggl[\exp\biggl\{-{\bigl(\sqrt{x\,e^{-A(t|t_0)}}-\sqrt{x_0}\bigr)^2\over R(t|t_0)}\biggr\} +\exp\biggl\{-{\bigl(\sqrt{x\,e^{-A(t|t_0)}}+\sqrt{x_0}\bigr)^2\over R(t|t_0)}\biggr\} \biggr],&\alpha(t)\neq 0, \end{array}\right. \end{eqnarray} (5.16)

    with \widetilde R(t|t_0) = \int_{t_0}^tr(\theta)\, d\theta and with A(t|t_0) and R(t|t_0) given in (4.2) and (5.3), respectively. We now take into account the cases 1. and 2.

    1. For a time-inhomogeneous Wiener process with A_1 = 0 and A_2(t) = r(t)/2 , for x_0 > 0 and x > 0 one has (cf. Giorno and Nobile [19])

    \begin{equation} a_X(\sqrt{x},t|\sqrt{x_0},t_0) = {1\over \sqrt{\pi\,\widetilde R(t|t_0) }}\biggl[\exp\biggl\{-{\bigl(\sqrt{x}-\sqrt{x_0}\bigr)^2\over \widetilde R(t|t_0)}\biggr\} -\exp\biggl\{-{\bigl(\sqrt{x}+\sqrt{x_0}\bigr)^2\over \widetilde R(t|t_0)}\biggr\}\biggr]. \end{equation} (5.17)

    Hence, Eq. (5.14) follows by comparing the first of (5.16) with (5.17).

    2. For a time-inhomogeneous Ornstein-Uhlenbeck process with B_1(x, t) = \alpha(t)\, x/2 and B_2(t) = r(t)/2 , one has (cf. Giorno and Nobile [19]):

    \begin{eqnarray} &&a_Y(\sqrt{x},t|\sqrt{x_0},t_0) = {e^{-A(t|t_0)/2}\over \sqrt{\pi R(t|t_0) }}\; \biggl[\exp\biggl\{-{\bigl(\sqrt{x\,e^{-A(t|t_0)}}-\sqrt{x_0}\bigr)^2\over R(t|t_0)}\biggr\} -\exp\biggl\{-{\bigl(\sqrt{x\,e^{-A(t|t_0)}}+\sqrt{x_0}\bigr)^2\over R(t|t_0)}\biggr\}\biggr],\\ &&x > 0,x_0 > 0. \end{eqnarray} (5.18)

    Then, by comparing the second of (5.16) with (5.18), Eq. (5.15) follows.

    We consider the TH-RF process Z(t) , obtained from (5.1) by setting \alpha(t) = \alpha , \beta(t) = \beta , r(t) = r , with \alpha\in\mathbb{R} , \beta > 0 , r > 0 and a zero-flux condition in the zero-state. From (2.6), for the TH-RF process Z(t) one has:

    \begin{equation} h_Z(x) = x^{-\beta/r}\exp\Bigl\{-{\alpha x\over r}\Bigr\}, \qquad s_Z(x) = {x^{\beta/r-1}\over r}\,\exp\Bigl\{{\alpha x\over r}\Bigr\}. \end{equation} (5.19)

    As proved by Feller [42], the boundary 0 is regular for 0 < \beta < r and entrance for \beta\geq r . Furthermore, the end-point +\infty is a nonattracting natural boundary for \alpha\leq 0 and an attracting natural boundary for \alpha > 0 .

    The transition pdf of the TH-RF process is obtainable from (5.4) by setting \alpha(t) = \alpha , \beta(t) = \beta , r(t) = r and \xi = \beta/r . In particular,

    ● if \alpha = 0 , one has:

    \begin{equation} r_Z(x,t|x_0) = \left\{\begin{array}{ll} { {1\over x\,\Gamma(\beta/r)}} \Bigl[{x\over r\,t}\Bigr]^{\beta/r}\,\exp\Bigl\{-{x\over r\,t}\Bigr\},&x_0 = 0,\\ \\ {1\over r\,t}\,\Bigl({x\over x_0}\Bigr)^{(\beta-r)/(2r)}\,\exp\Bigl\{-{x_0+x\over r\,t}\Bigr\} \,I_{\beta/r-1}\Bigl[ {2\sqrt{x\,x_0}\over r\,t}\Bigr], &x_0 > 0, \end{array}\right. \end{equation} (5.20)

    ● if \alpha\neq 0 , one obtains:

    \begin{equation} r_Z(x,t|x_0) = \left\{\begin{array}{ll} {{1\over x\,\Gamma(\beta/r)}}\Bigl[ {\alpha\, x\over r(e^{\alpha\,t}-1)}\Bigr]^{\beta/r}\exp\Bigl\{-{\alpha\,x\over r(e^{\alpha\,t}-1)}\Bigr\} ,&x_0 = 0,\\ \\ {\alpha\over r(e^{\alpha\,t}-1)}\,\Bigl[ {x\,e^{-\alpha\,t}\over x_0}\Bigr]^{(\beta-r)/(2r)}\exp\Bigl\{-{\alpha\,[x+x_0\,e^{\alpha\,t}]\over r(e^{\alpha\,t}-1)}\Bigr\}\, I_{\beta/r-1}\Bigl[ {2\alpha\sqrt{x\,x_0\,e^{\alpha\,t}}\over r(e^{\alpha\,t}-1)}\Bigr], &x_0 > 0. \end{array}\right. \end{equation} (5.21)

    The conditional mean and the conditional variance of the TH-RF process can be obtained from (5.5) with \xi = \beta/r . We note that if \alpha < 0 , \beta > 0 and r > 0 , the TH-RF process admits the steady-state density given in (5.7) with \xi = \beta/r .

    In Figure 6, we consider a TH-RF process Z(t) in [0, +\infty) , having C_1(x) = \alpha\, x+\beta and C_2(x) = 2\, r\, x , with \alpha = 0 , \beta = 0.25 , r = 0.5 , x_0 = 5 and t_0 = 0 . We compare the histogram of the random sample of N = 5\cdot 10^4 observations of Z(t) with the transition pdf (5.20) as function of x (x\geq 0) for t = 0.5 on the left and t = 1 on the right. Instead, in Figure 7, we consider a TH-RF process Z(t) in [0, +\infty) , with \alpha = -0.5 , \beta = 0.25 , r = 0.5 , x_0 = 5 and t_0 = 0 and we compare the histogram of the random sample of N = 5\cdot 10^4 observations of Z(t) with the transition pdf (5.21) as function of x (x\geq 0) for t = 0.5 on the left and t = 1 on the right. Figures 6 and 7 show the good agreement between the exact and simulated results.

    Figure 6.  For the TH-RF process Z(t) , with \alpha = 0 , \beta = 0.25 , r = 0.5 , x_0 = 5 , the transition pdf (5.20) is plotted (red curve) as function of x , for t = 0.5 (on the left) and t = 1 (on the right) and the histogram, obtained via the simulation of the sample paths, is superimposed over the density.
    Figure 7.  As in Figure 6 with \alpha = -0.5 , \beta = 0.25 , r = 0.5 and x_0 = 5 .

    In Giorno and Nobile [43] the FPT problem through a state S starting from x_0 for the TH-RF process has been considered. When 0 < S < x_0 the ultimate FPT probability P_Z(S|x_0) = 1 if and only if [\alpha < 0, \beta > 0] or [\alpha = 0, 0 < \beta\leq r] . Making use of (2.7), for \alpha = 0 and 0 < \beta\leq r one has that t_1^{(Z)}(S|x_0) diverges, whereas if \alpha < 0 and \beta > 0 one obtains:

    \begin{equation} t_1^{(Z)}(S|x_0) = {1\over |\alpha|}\, \Gamma\Bigl({\beta\over r}\Bigr)\int_{|\alpha|S/r}^{|\alpha|x_0/r}z^{-\beta/r}\,e^z\;dz -{1\over\beta}\,\sum\limits_{n = 0}^{+\infty}{1\over (1+\beta/r)_n}\,\Bigl({|\alpha|\over r}\Bigr)^n\;{x_0^{n+1}-S^{n+1}\over n+1},\quad 0 < S < x_0. \end{equation} (5.22)

    Moreover, if x_0 > 0 the ultimate FPT probability P_Z(0|x_0) = 1 if and only if \alpha\leq 0 and 0 < \beta < r . From (2.7), for \alpha = 0 and 0 < \beta < r one has that t_1^{(Z)}(0|x_0) diverges, whereas for \alpha < 0 and 0 < \beta < r (5.22) holds with S = 0 .

    Instead, when 0\leq x_0 < S , one obtains P_Z(S|x_0) = 1 and from (2.8) it results

    \begin{equation} t_1^{(Z)}(S|x_0) = {1\over\beta}\,\int_{x_0}^S \Phi\Bigl( 1,{\beta\over r}+1;- {\alpha\,z\over r}\Bigr)\;dz = {1\over\beta}\,\sum\limits_{n = 0}^{+\infty}{1\over (1+\beta/r)_n}\,\Bigl(-{\alpha\over r}\Bigr)^n\,{S^{n+1}-x_0^{n+1}\over n+1},\quad 0\leq x_0 < S, \end{equation} (5.23)

    so that for \alpha = 0 one has t_1^{(Z)}(S|x_0) = (S-x_0)/\beta . Moreover, when 0\leq x_0 < S , from (5.23) it follows:

    t_1^{(Z)}(S|x_0) decreases as \beta increases and

    \lim\limits_{\beta\to 0} t_1^{(Z)}(S|x_0) = +\infty, \qquad \lim\limits_{\beta\to+\infty} t_1^{(Z)}(S|x_0) = 0;

    t_1^{(Z)}(S|x_0) decreases as r increases and

    \lim\limits_{r\to 0} t_1^{(Z)}(S|x_0) = \left\{\begin{array}{ll} +\infty,&\alpha < 0,\\ {S-x_0\over\beta},&\alpha = 0,\\ {1\over\alpha}\ln\Bigl({\alpha\,S+\beta\over\alpha\,x_0+\beta}\Bigr),&\alpha > 0, \end{array}\right. \qquad \lim\limits_{r\to+\infty} t_1^{(Z)}(S|x_0) = {S-x_0\over \beta}\cdot

    In Figures 8 and 9, the FPT mean (5.23) of the TH-RF process is plotted for x_0 = 5 , S = 10 and for different choices of \beta and r , with \alpha = -0.02, 0.02 , respectively.

    Figure 8.  The FPT mean (5.23) is plotted for \alpha = -0.02 , x_0 = 5 and S = 10 as function of \beta on the left and as function of r on the right.
    Figure 9.  As in Figure 8 with \alpha = 0.02 .

    Making use of (5.19) in Remark 2.1, one can derive the following asymptotic results for the FPT moments of the TH-RF process.

    Remark 5.1. For the TH-RF process Z(t) , when n = 1, 2, \ldots one has:

    1. {\lim\limits_{S\uparrow+\infty}{ t_n^{(Z)}(S|0)\over [t_1^{(Z)}(S|0)]^n} = n!}, \qquad \alpha < 0,

    2. {\lim\limits_{S\downarrow 0}{ t_n^{(Z)}(S|0)\over [t_1^{(Z)}(S|0)]^n} = u_n}, \qquad 0 < \beta < r

    where

    u_0 = 1,\quad u_n = \sum\limits_{k = 1}^n {n\choose k} {(-1)^{k-1}\over \prod_{i = 0}^{k-1}\Bigl(1+i\,{r\over\beta}\Bigr)}\;u_{n-k},\quad n = 1,2,\ldots

    From Remark 5.1, the following asymptotic behaviors hold:

    ● for S\uparrow +\infty one has

    t_n^{(Z)}(S|0)\simeq n!\, [t_1^{(Z)}(S|0)]^n,\qquad \alpha < 0,

    so that for \alpha < 0 the FPT density of TH-RF process exhibits an exponential trend for large boundary;

    ● for S\downarrow 0 it results:

    t_n^{(Z)}(S|0)\simeq u_n \,[t_1^{(Z)}(S|0)]^n = m_n^{(Z)}(S|0), \qquad 0 < \beta < r.

    In particular, one has:

    m_2^{(Z)}(S|0) = {\beta+2r\over \beta+r}\,[t_1^{(Z)}(S|0)]^2,\quad m_3^{(Z)}(S|0) = {\beta^2+6r\beta+12r^2\over (\beta+r)(\beta+2r)}\,[t_1^{(Z)}(S|0)]^3.

    In Tables 1011, t_1^{(Z)}(S|0) is computed via (5.23), whereas t_2^{(Z)}(S|0) and t_3^{(Z)}(S|0) are numerically evaluated making use of (2.8) and (5.19). In Table 10, the mean t_1^{(Z)}(S|0) , the variance {\rm Var}^{(Z)}(S|0) , the coefficient of variation {\rm Cv}^{(Z)}(S|0) and the skewness \Sigma^{(Z)}(S|0) of the FPT are listed for \alpha = -0.02 , \beta = 0.1 , r = 0.5 and some choices of S > 0 . Being \alpha < 0 , the coefficient of variation and the skewness approach to 1 and to 2, respectively, as S increases.

    Table 10.  For the TH-RF process, with \alpha = -0.02 , \beta = 0.1 and r = 0.5 , t_1^{(Z)}(S|0) , {\rm Var}^{(Z)}(S|0) , {\rm Cv}^{(Z)}(S|0) and \Sigma^{(Z)}(S|0) are listed for increasing values of S > 0 .
    S t_1^{(Z)}(S|0) {\rm Var}^{(Z)}(S|0) {\rm Cv}^{(Y)}(Z|0) \Sigma^{(Z)}(S|0)
    10. 118.892 12127.6 0.926266 1.99421
    20. 286.387 72259.4 0.938630 1.99609
    30. 524.198 247872. 0.949771 1.99746
    40. 864.044 687422. 0.959568 1.99839
    50. 1352.34 1.71357\times 10^6 0.967980 1.99902
    60. 2057.09 4.02295\times 10^6 0.975033 1.99942
    70. 3078.07 9.11448\times 10^6 0.980817 1.99966
    80. 4561.82 2.02095\times 10^7 0.985460 1.99981
    90. 6723.80 4.42308\times 10^7 0.989117 1.99988
    100. 9881.02 9.60682\times 10^7 0.991946 1.99993
    300. 2.31543\times 10^7 5.36117\times 10^{14} 0.999993 1.99952
    600. 3.24856\times 10^{12} 1.05531\times 10^{25} 1.0 2.00237
    900. 4.86065\times 10^{17} 2.36259\times 10^{35} 1.0 1.99494
    1200. 7.45766\times 10^{22} 5.56166\times 10^{45} 1.0 2.00006

     | Show Table
    DownLoad: CSV
    Table 11.  For the TH-RF process with \alpha = -0.02 , \beta = 0.1 and r = 0.5 , t_1^{(Z)}(S|0) , t_2^{(Z)}(S|0) and t_3^{(Z)}(S|0) and their approximate values m_2^{(Z)}(S|0) and m_3^{(Z)}(S|0) are listed for decreasing values of S .
    S t_1^{(Z)}(S|0) t_2^{(Z)}(S|0) m_2^{(Z)}(S|0) t_3^{(Z)}(S|0) m_3^{(Z)}(S|0)
    3.0 3.155612\times 10^{1} 1.833106\times 10^{3} 1.825612\times 10^{3} 1.589679\times 10^{5} 1.575922\times 10^{5}
    2.5 2.607399\times 10^{1} 1.250665\times 10^{3} 1.246397\times 10^{3} 8.954832\times 10^{4} 8.890094\times 10^{4}
    2.0 2.068314\times 10^{1} 7.864368\times 10^{2} 7.842855\times 10^{2} 4.463324\times 10^{4} 4.437447\times 10^{4}
    1.5 1.538192\times 10^{1} 4.346661\times 10^{2} 4.337727\times 10^{2} 1.833211\times 10^{4} 1.825220\times 10^{4}
    1.0 1.016871\times10^{1} 1.898320\times 10^{2} 1.895714\times 10^{2} 5.288689\times 10^{3} 5.273284\times 10^{3}
    0.9 9.136485 1.532275\times 10^{2} 1.530382\times 10^{2} 3.834971\times 10^{3} 3.824913\times 10^{3}
    0.8 8.107709 1.206466\times 10^{2} 1.205141\times 10^{2} 2.679123\times 10^{3} 2.672874\times 10^{3}
    0.7 7.082364 9.204832 \times 10^{1} 9.195978\times 10^{1} 1.785281\times 10^{3} 1.781635\times 10^{3}
    0.6 6.060439 6.739193\times 10^{1} 6.733635\times 10^{1} 1.118298\times 10^{3} 1.116339\times 10^{3}
    0.5 5.041920 4.663716\times 10^{1} 4.660510\times 10^{1} 6.437341\times 10^{2} 6.427943\times 10^{2}
    0.4 4.026796 2.974403\times 10^{1} 2.972766\times 10^{1} 3.278467\times 10^{2} 3.274636\times 10^{2}
    0.3 3.015055 1.667290\times 10^{1} 1.666602\times 10^{1} 1.375785\times 10^{2} 1.374579\times 10^{2}
    0.2 2.006683 7.384456 7.382423 4.054846\times 10^{1} 4.052474\times 10^{1}
    0.1 1.001669 1.839710 1.839457 5.041775 5.040300

     | Show Table
    DownLoad: CSV

    Moreover, in Table 11, for the TH-RF process with \alpha = -0.02 , \beta = 0.1 and r = 0.5 , we compare the FPT moments t_2^{(Z)}(S|0) and t_3^{(Z)}(S|0) with the approximate values m_2^{(Z)}(S|0) and m_3^{(Z)}(S|0) , respectively. We note that the goodness of the approximations improves as S approaches zero.

    The TNH-RW, TNH-ROU and TNH-RF diffusion processes can be seen as the continuous approximations of some time-inhomogeneous birth-death processes that modeling queueing systems in heavy-traffic conditions. Referring to the queueing systems, in this section we consider some examples in which the infinitesimal drifts and the infinitesimal variances are time-dependent and include periodic functions. The presence of periodicity in the infinitesimal moments express the existence of rush hours occurring on a daily basis. For the considered reflected processes, we apply the exact and asymptotic results obtained in Sections 3–5 to analyze the conditional averages and the conditional variances for some choices of the periodic functions and of the parameters.

    Example 6.1. (Wiener model) We consider the TNH-RW process X(t) , having infinitesimal drift and infinitesimal variance A_1(t) = \gamma\, \sigma^2(t) and A_2(t) = \sigma^2(t) , with \gamma\in\mathbb{R} and \sigma(t) > 0 . This process can be seen as the continuous approximation of the birth-death queueing system N_1(t) with arrival and departure intensity functions \lambda_n(t) = \lambda\sigma^2(t)/\varepsilon+\sigma^2(t)/(2\varepsilon^2) (n = 0, 1, \ldots) and \mu_n(t) = \mu\sigma^2(t)/\varepsilon+\sigma^2(t)/(2\varepsilon^2) (n = 1, 2, \ldots) , where \lambda > 0 , \mu > 0 and \varepsilon is a positive scaling parameter. Indeed, the scaled process N_1(t)\, \varepsilon converges weakly to the diffusion process X(t) , having state-space [0, +\infty) , with infinitesimal moments (see, for instance, [1]):

    A_1(t) = \lim\limits_{\varepsilon\downarrow 0} \varepsilon\,\Bigl[\lambda_n(t)-\mu_n(t)\Bigr] = \gamma\,\sigma^2(t),\qquad A_2(t) = \lim\limits_{\varepsilon\downarrow 0} \varepsilon^2\,\Bigl[\lambda_n(t)+\mu_n(t)\Bigr] = \sigma^2(t),

    with \gamma = \lambda-\mu . We assume that

    \begin{equation} \sigma^2(t) = \nu\,\biggl[1+c\,\sin\Bigl({2\pi\,t\over Q}\Bigr)\biggr],\qquad t\geq 0, \end{equation} (6.1)

    where \nu > 0 is the average of the period function \sigma^2(t) of period Q and c is the amplitude of the oscillations, with 0\leq c < 1 . These choices of parameters ensure that the infinitesimal variance is a positive function.

    In Figure 10, we suppose that \sigma^2(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})] and that at the initial time t_0 = 0 the number of customers is X(t_0) = x_0 = 5 . As proved in Corollary 3.2, the transition pdf of X(t) admits the asymptotic exponential behavior (3.9) for \gamma < 0 . In Figure 10, the conditional mean and the conditional variance, obtained from (3.7), and the related asymptotic behaviors are shown as function of t for \gamma = -0.5, -1 ; the dotted lines indicate the corresponding asymptotic means and variances (2\, |\gamma|)^{-1} and (2\, |\gamma|)^{-2} . $

    Figure 10.  For the TNH-RW process, the conditional mean and the conditional variance are plotted as function of t for t_0 = 0 , x_0 = 5 , \sigma^2(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})] ; the dotted lines indicate the corresponding asymptotic means and variances.

    Example 6.2. (Ornstein-Uhlenbeck model) We consider the TNH-ROU process Y(t) , having infinitesimal drift and infinitesimal variance B_1(x, t) = \alpha\, x and B_2(t) = \sigma^2(t) , with \alpha\in\mathbb{R} and \sigma(t) > 0 . This process can be seen as the continuous approximation of the birth-death queueing system N_2(t) with arrival and departure intensity functions \lambda_n(t) = \lambda\, n+\sigma^2(t)/(2\varepsilon^2) (n = 0, 1, \ldots) and \mu_n(t) = \mu\, n+\sigma^2(t)/(2\varepsilon^2) (n = 1, 2, \ldots) , both depending on the number of customers in the systems, where \lambda > 0 , \mu > 0 and \varepsilon is a positive scaling parameter. Indeed, the scaled process N_2(t)\, \varepsilon converges weakly to the diffusion process X(t) , having state-space [0, +\infty) , with infinitesimal moments (see, for instance, [1]):

    B_1(x,t) = \lim\limits_{\varepsilon\downarrow 0\atop n = x/\varepsilon} \varepsilon\,\Bigl[\lambda_n(t)-\mu_n(t)\Bigr] = \alpha\,x,\qquad B_2(t) = \lim\limits_{\varepsilon\downarrow 0\atop n = x/\varepsilon} \varepsilon^2\,\Bigl[\lambda_n(t)+\mu_n(t)\Bigr] = \sigma^2(t),

    where \alpha = \lambda-\mu . We assume that (6.1) holds.

    As proved in Corollary 4.2, when \alpha < 0 the transition pdf of Y(t) admits the asymptotic behavior (4.11) with

    \omega_1(t) = {\nu\over 2\,|\alpha|}\, \biggl\{1+{c\,Q\,\alpha\over \pi^2+Q^2\alpha^2}\Bigr[\pi\,\cos\Bigl( {2\pi\,t\over Q} \Bigr)+Q\,\alpha\,\sin\Bigl( {2\pi\,t\over Q} \Bigr)\Bigr]\biggr\}, \qquad t\geq 0.

    In Figure 11, the conditional mean and the conditional variance, obtained from (4.8), and the related asymptotic behaviors are shown as function of t for \sigma^2(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})] , t_0 = 0 , x_0 = 5 and \alpha = -0.05, -0.1 ; the dotted functions indicate the corresponding asymptotic mean [2\omega_1(t)/\pi]^{1/2} and asymptotic variances \omega_1(t)\, (1-2/\pi) . $

    Figure 11.  For the TNH-ROU process, the conditional mean and the conditional variance are plotted as function of t for t_0 = 0 , x_0 = 5 , \sigma^2(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})] ; the dotted functions indicate the corresponding asymptotic means and variances.

    Example 6.3. (Feller model) We consider the TNH-RF process Z(t) , having infinitesimal drift and infinitesimal variance C_1(x, t) = \alpha\, x+\xi\, r(t) and C_2(x, t) = 2\, r(t)\, x , with \alpha\in\mathbb{R} , \xi > 0 and r(t) > 0 . This process can be seen as the continuous approximation of the birth-death queueing system N_3(t) with arrival and departure intensity functions \lambda_n(t) = [\lambda+r(t)/\varepsilon]\, n+\xi\, r(t)/\varepsilon (n = 0, 1, \ldots) and \mu_n(t) = [\mu+r(t)/\varepsilon]\, n (n = 1, 2, \ldots) , both depending on the number of customers in the systems and on the positive scaling parameter \varepsilon . Indeed, the scaled process N_3(t)\, \varepsilon converges weakly to a diffusion process, having state-space [0, +\infty) (see, for instance, [34]) and one has:

    C_1(x,t) = \lim\limits_{\varepsilon\downarrow 0\atop n = x/\varepsilon} \varepsilon\,\Bigl[\lambda_n(t)-\mu_n(t)\Bigr] = \alpha\,x+\xi\,r(t),\qquad C_2(x,t) = \lim\limits_{\varepsilon\downarrow 0\atop n = x/\varepsilon} \varepsilon^2\,\Bigl[\lambda_n(t)+\mu_n(t)\Bigr] = 2\,r(t)\,x,

    where \alpha = \lambda-\mu . We assume that r(t) = \sigma^2(t) , with \sigma^2(t) given in (6.1).

    As proved in Corollary 5.1, when \alpha < 0 the transition pdf of Z(t) admits the asymptotic behavior (5.8) with

    \omega_2(t) = {1\over \nu}\,\biggl\{ {1\over |\alpha|}-{c\,Q\over 4\pi^2+Q^2\,\alpha^2}\,\biggl[2\pi \cos\Bigl({2\pi t\over Q}\Bigr)\\ +\alpha\, Q\sin\Bigl({2\pi t\over Q}\Bigr)\biggr]\biggr\}^{-1},\qquad t\geq 0.

    In Figure 12, the conditional mean and the conditional variance, obtained from (5.5), and the related asymptotic behaviors are shown as function of t for r(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})] , t_0 = 0 , x_0 = 5 , \alpha = -0.08 and \xi = 1.0, 2.0 ; the dotted functions indicate the corresponding asymptotic mean \xi/\omega_2(t) and asymptotic variances \xi/[\omega_2(t)]^2 . $

    Figure 12.  For the TNH-RF process, the conditional mean and the conditional variance are plotted as function of t for t_0 = 0 , x_0 = 5 , \alpha = -0.08 , r(t) = 0.5\, [1+0.9\, \sin({2\pi\, t/5})] ; the dotted functions indicate the corresponding asymptotic means and variances.

    As highlighted in the Figures 1012, the periodic intensity function (6.1), used in the infinitesimal drifts and in the infinitesimal variances of the reflected Wiener, Ornstein-Uhlenbeck and Feller models, affects the shapes of the conditional averages and of the conditional variances.

    For the Wiener, Ornstein Uhlenbeck and Feller processes, restricted to the interval [0, +\infty) , with reflecting or a zero-flux condition in the zero-state, we analyze the transition probability density functions and their asymptotic behaviors, paying particular attention to the time-inhomogeneous proportional cases and to the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Moreover, the FPT moments and their asymptotic behaviors are analyzed for the time-homogeneous cases. Finally, some applications of the obtained results to queueing systems are considered. Various numerical computations are performed with Mathematica to illustrate the role played by parameters.

    Let f_Y(x, t|x_0) be the transition pdf of the unrestricted time-homogeneous Ornstein-Uhlenbeck process with infinitesimal moments B_1(x) = \alpha x+\beta and B_2 = \sigma^2 . For x, x_0\in{\mathbb R} , we denote by

    \begin{equation} j_Y(x,t|x_0) = (\alpha\, x+\beta)\, f_Y(x,t|x_0)-{\sigma^2\over 2} \,{\partial\over\partial x} f_Y(x,t|x_0) \end{equation} (A.1)

    the probability current of the unrestricted Ornstein-Uhlenbeck process. The transition pdf r_Y(x, t|x_0) of a time-homogeneous diffusion process restricted to [0, +\infty) , with zero reflecting boundary, satisfies the following integral equations (cf. Giorno et al. [44]):

    \begin{equation} \begin{split} &r_Y(x,t|0) = 2\, f_Y(x,t|0)-2\int_0^t j_Y(0,\tau | 0)\,r_Y(x,t-\tau|0)\;d\tau,\quad x\geq 0, \\ &r_Y(x,t|x_0) = f_Y(x,t|x_0)-\int_0^t j_Y(0,\tau |x_0)\,r_Y(x,t-\tau|0)\;d\tau, \;\; x\geq 0,x_0 > 0. \end{split} \end{equation} (A.2)

    Taking the LT in (A.2) one has:

    \begin{equation} r_{\lambda}^{(Y)}(x|0) = {2\,f_{\lambda}^{(Y)}(x|0)\over 1+2\,j_{\lambda}^{(Y)}(0|0)},\qquad r_{\lambda}^{(Y)}(x|x_0) = f_{\lambda}^{(Y)}(x|x_0)-j_{\lambda}^{(Y)}(0|x_0)\,r_{\lambda}^{(Y)}(x|0), \end{equation} (A.3)

    where f_{\lambda}^{(Y)}(x|x_0) is LT of f_Y(x, t|x_0) and j_{\lambda}^{(Y)}(0|x_0) is LT of j_Y(0, t|x_0) . Taking the LT in the first equation in (4.4), with \alpha(t) = \alpha , \beta(t) = \beta and \sigma^2(t) = \sigma^2 , we have:

    \begin{equation} f_{\lambda}^{(Y)}(x|x_0) = \left\{\begin{array}{ll} {2^{{\lambda\over |\alpha|}-1}\over \sigma\pi\sqrt{|\alpha|}} \Gamma\Bigl( {\lambda\over 2 |\alpha|} \Bigr)\, \Gamma\Bigl({1\over 2}+{\lambda\over 2|\alpha|}\Bigr) \exp\Bigl\{-{|\alpha|\over 2\,\sigma^2}\Bigl[\Bigl(x+{\beta\over\alpha}\Bigr)^2-\Bigl(x_0+{\beta\over\alpha}\Bigr)^2\Bigr]\Bigr\}\\ \quad \times D_{-{\lambda\over |\alpha|}}\Bigl(-{\sqrt{2|\alpha|}\over \sigma}\Bigl[x_0\wedge x+{\beta\over\alpha}\Bigr]\Bigr) D_{-{\lambda\over |\alpha|}}\Bigl({\sqrt{2|\alpha|}\over \sigma}\Bigl[x_0\vee x+{\beta\over\alpha}\Bigr]\Bigr),&\alpha < 0,\\ \\ {2^{{\lambda\over\alpha}}\over \sigma\pi\sqrt{\alpha}} \Gamma\Bigl( 1+{\lambda\over 2 \alpha}\Bigr)\, \Gamma\Bigl({1\over 2}+{\lambda\over 2\alpha}\Bigr) \exp\Bigl\{-{\alpha\over 2\,\sigma^2}\Bigl[\Bigl(x_0+{\beta\over\alpha}\Bigr)^2-\Bigl(x+{\beta\over\alpha}\Bigr)^2\Bigr]\Bigr\}\\ \quad \times D_{-{\lambda\over \alpha}-1}\Bigl(-{\sqrt{2\alpha}\over \sigma}\Bigl[x_0\wedge x+{\beta\over\alpha}\Bigr]\Bigr) D_{-{\lambda\over \alpha}-1}\Bigl({\sqrt{2\alpha}\over \sigma}\Bigl[x_0\vee x+{\beta\over\alpha}\Bigr]\Bigr),&\alpha > 0, \end{array}\right. \end{equation} (A.4)

    with x_0\wedge x = \min(x_0, x) and x_0\vee x = \max(x_0, x) . Moreover, taking the LT in Eq. (A.1) one has:

    \begin{equation} j_{\lambda}^{(Y)}(0|x_0) = \left\{\begin{array}{ll} {1\over 2}\,{\cal I}(x_0)-{2^{{\lambda\over |\alpha|}-{1\over2}}\over \pi } \exp\Bigl\{{|\alpha|\over 2\,\sigma^2}\Bigl[\Bigl(x_0+{\beta\over\alpha}\Bigr)^2-\Bigl({\beta\over\alpha}\Bigr)^2\Bigr]\Bigr\}\, \Gamma\Bigl( {1\over 2}+{\lambda\over 2 |\alpha|} \Bigr)\\ \qquad\times\Gamma\Bigl( 1+{\lambda\over 2 |\alpha|} \Bigr)\, D_{-{\lambda\over |\alpha|}}\Bigl({\sqrt{2|\alpha|}\over \sigma}\Big(x_0+{\beta\over\alpha}\Bigr)\Bigr) D_{-{\lambda\over |\alpha|}-1}\Bigl(-{\sqrt{2|\alpha|}\over \sigma}{\beta\over\alpha}\Bigr),&\alpha < 0\\ \\ {1\over 2}\,{\cal I}(x_0)-{2^{{\lambda\over \alpha}-{1\over 2}}\over \pi } \exp\Bigl\{-{\alpha\over 2\,\sigma^2}\Bigl[\Bigl(x_0+{\beta\over\alpha}\Bigr)^2-\Bigl({\beta\over\alpha}\Bigr)^2\Bigr]\Bigr\}\, \Gamma\Bigl( {1\over 2}+{\lambda\over 2 \alpha} \Bigr)\\ \qquad\times\Gamma\Bigl( 1+{\lambda\over 2 \alpha} \Bigr)\, D_{-{\lambda\over \alpha}-1}\Bigl({\sqrt{2\alpha}\over \sigma}\Big(x_0+{\beta\over\alpha}\Bigr)\Bigr) D_{-{\lambda\over \alpha}}\Bigl(-{\sqrt{2\alpha}\over \sigma}{\beta\over\alpha}\Bigr),&\alpha > 0\\ \end{array}\right. \end{equation} (A.5)

    where {\cal I}(x_0) = 1 for x_0 = 0 and {\cal I}(x_0) = 0 for x_0 > 0 . Therefore, making use of (A.4) and (A.5) in (A.3) and recalling the following relation

    D_{\nu-1}(z)\,D_{\nu}(-z)+D_{\nu}(z)\,D_{\nu-1}(-z) = {\pi\,2^{\nu+1/2}\over \Gamma\Bigl(1-{\nu\over 2}\Bigr)\,\Gamma\Bigl({1\over 2}-{\nu\over 2}\Bigr)},

    one obtains (4.13).

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This research is partially supported by MIUR - PRIN 2017, project "Stochastic Models for Complex Systems", no. 2017JFFHSH. The authors are members of the research group GNCS of INdAM.

    The authors declare there is no conflict of interest.



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