[]F. Cinque []A Note on the Conditional Probabilities of the Telegraph Process
[] []

# A Note on the Conditional Probabilities of the Telegraph Process111Submitted
to Statistics and Probability Letters; version of February 3, 2022

Fabrizio Cinque
Department of Statistical Sciences, Sapienza University of Rome, Italy,
<EMAIL_ADDRESS>
###### Abstract

We consider the telegraph process with two velocities,
$a_{1}>a_{2}\in\mathbb{R}$, and two rates of reversal,
$\lambda_{1},\lambda_{2}>0$. We study some of its features with respect to the
conditional probability measure where both the initial speed and the number of
changes of direction are known. We exhibit a new proof by induction of the
(conditional) probability law and a detailed study of the distribution of the
motion at time $t>0$ conditioned on its position at a previous time $0<s<t$.
In the case of a symmetric process, we present some results on the joint
distribution of the position of the motion at time $t>0$, its maximum and its
minimum up to that moment.

Keywords:  Random Motions with Finite Velocities; Motions with Reflections;
Stochastic Motions with Drift; Distribution of Maximum and Minimum; Induction
Principle; Reflection Principle

2020 MSC:  Primary 60K99; 60G50

##  1 Introduction

Let $V_{0}$ be a uniformly distributed random variable taking values
$a_{1},a_{2}\in\mathbb{R}$, with $a_{1}>a_{2}$. Let $N=\\{N(t)\\}_{t\geq 0}$
be a non decreasing counting process. We define the random velocity process
$\\{V(t)\\}_{t\geq 0}$ as

$V(t):=\frac{a_{1}+a_{2}}{2}+\Bigl{(}V_{0}-\frac{a_{1}+a_{2}}{2}\Bigr{)}(-1)^{N(t)},$
(1.1)

thus $V(t)\in\\{a_{1},a_{2}\\}\ a.s.$ and $V(t)=V(0)=V_{0}$ if and only if
$N(t)$ is even. The telegraph process $\mathcal{T}=\\{\mathcal{T}(t)\\}_{t\geq
0}$ is defined as

$\mathcal{T}(t):=\int_{0}^{t}V(s)\mathop{}\\!\mathrm{d}s=\sum_{i=0}^{N(t)-1}\Bigl{(}T_{i+1}-T_{i}\Bigr{)}V(T_{i})+\Bigl{(}t-T_{N(t)}\Bigr{)}V(T_{N(t)}),$
(1.2)

where $T_{n},\ n\in\mathbb{N}$, is the $n$-th arrival time of $N$, $T_{0}=0\
a.s.$ and $V(T_{n})$ is the velocity after the $n$-th change of direction.
$\mathcal{T}(t)$ represents the position, at time $t$, of a particle which
starts moving with velocity $V(0)=V_{0}$ at time $t=0$ and changes its speed
when an event of the point process $N$ occurs. As usual, we assume that $N$ is
a Poisson-type process with independent and exponentially distributed waiting
times with two alternating parameters, $\lambda_{1}>0$ if $V(t)=a_{1}$ and
$\lambda_{2}>0$ if $V(t)=a_{2}$. Therefore, the telegraph motion proceeds with
velocity $a_{1}$ ($a_{2}$) for an exponential random time of average
$1/\lambda_{1}$ ($1/\lambda_{2}$). $\mathcal{T}(t)$ is in
$\\{a_{1}t,a_{2}t\\}$ if no changes of direction occur in $[0,t]$. Thus,
$P\\{\mathcal{T}(t)=a_{i}t\\}=P\\{V(0)=a_{i},N(t)=0\\}=e^{-\lambda_{i}t}/2$.
Throughout the paper, we call $\mathcal{T}$ asymmetric telegraph process if
$N$ is a homogeneous Poisson process ($\lambda_{1}=\lambda_{2}$). If there is
also $a_{1}=-a_{2}>0$, then, we call $\mathcal{T}$ symmetric telegraph
process.

The (symmetric) telegraph process has been introduced by Goldstein (1951). Its
transition density satisfies the following partial differential equation
(which, in the symmetric case, $\lambda_{1}=\lambda_{2},\ a_{1}=-a_{2}>0$,
reduces to the telegraph equation)

$\frac{\partial^{2}p}{\partial
t^{2}}+(a_{1}+a_{2})\frac{\partial^{2}p}{\partial t\partial
x}+(\lambda_{1}+\lambda_{2})\frac{\partial p}{\partial
t}=-a_{1}a_{2}\frac{\partial^{2}p}{\partial
x^{2}}+\frac{(a_{1}+a_{2})(\lambda_{1}+\lambda_{2})+(a_{1}-a_{2})(\lambda_{1}-\lambda_{2})}{2}\frac{\partial
p}{\partial x}.$ (1.3)

For $a_{1}>0>a_{2}$, under the generalized Kac’s conditions
$\lambda_{1},\lambda_{2},a_{1},-a_{2}\longrightarrow\infty$ such that
$\lambda_{1}/\lambda_{2}\longrightarrow 1,\
-a_{1}a_{2}/(\lambda_{1}+\lambda_{2})\longrightarrow\sigma^{2}/2>0$ (or
alternatively
$-a_{1}a_{2}/\sqrt{\lambda_{1}\lambda_{2}}\longrightarrow\sigma^{2}$) and
$(a_{1}\lambda_{1}+a_{2}\lambda_{2})/(\lambda_{1}+\lambda_{2})\longrightarrow-\mu\in\mathbb{R}$,
equation (1.3) converges to the heat equation with drift $\mu$ and diffusivity
$\sigma^{2}/2$ and the telegraph process converges in distribution to standard
Brownian motion with drift $\mu$ and diffusion $\sigma^{2}$. Several authors
used this connection with the partial differential equations to obtain some
important results, see for instance Kac (1974), Orsingher (1990) and Kolesnik
(1998). The distribution of the telegraph process with two different rates and
velocities has been first proved in Beghin et al. (2001) by means of two
different approaches, one of which based on the governing partial differential
equation (1.3). Among others, some remarkable results were proved by Di
Crescenzo (2001), Stadje and Zacks (2004), Zacks (2004) and more recently by
Di Crescenzo et al. (2013) and Lopez and Ratanov (2014). In the latter, the
authors show also the explicit probabilities of the telegraph process
conditioned on both the starting velocity and the number of changes of
direction. In the second section of this paper, we show an alternative proof
of these distributions.

The telegraph motion has a finite velocity and it is suitable to describe real
motions that emerge in different fields, like geology, see Travaglino et al.
(2018), finance, see Di Crescenzo and Pellerey (2002), Ratanov (2007),
Kolesnik and Ratanov (2013), and physics, see Hartmann et al. (2020).

The telegraph process is non-Markov, while the couple
$\\{\mathcal{T}(t),V(t)\\}_{t\geq 0}$ is it. This fact can be found in Davis
(1984) and it has been used to infer the rate parameter of the motion, see De
Gregorio and Iacus (2008). In Section 3, we display the exact distribution of
$\mathcal{T}(t)$ conditioned on the position at previous time
$\mathcal{T}(s)=x$ with $0\leq s<t$ and $x\in[a_{2}s,a_{1}s]$. This
conditional law may be very useful for the parametric estimation of the
motion, for instance, by applying it in a pairwise composite likelihood.

At last, in the fourth section we study the conditional joint distribution of
the symmetric telegraph process, its minimum and its maximum. We provide a
general recurrent formula and explicit results under some particular condition
which also lead to a reflection-type principle resembling the negative
reflection principle, introduced in Cinque (2020), concerning finite velocity
random motions. We recall that several papers have been devoted to the
analysis of the first passage times, see Foong (1992), Foong and Kanno (1994),
Orsingher (1995) and more recently of Lopez and Ratanov (2014), Mori et al.
(2020), De Bruyne et al. (2021) and Ratanov (2021). The investigation of the
motion in presence of multiple boundaries has been recently carried out by Di
Crescenzo et al. (2018) and Di Crescenzo et al. (2020). On the other hand, the
maximum has been further studied in Cinque and Orsingher (2020, 2021) and
Cinque (2020).

##  2 Conditional probability law of the telegraph process

In this section we show a new induction approach to study the probability of
the telegraph process $\mathcal{T}$, given the initial velocity and the number
of changes of direction. This method permits us to obtain explicit conditional
probabilities concerning the sum of the odd waiting times,
$T_{2k+1}-T_{2k},k\in\mathbb{N}_{0},$ of the Poisson-type process
$N=\\{N(t)\\}_{t\geq 0}$, governing the switches of the random motion. It is
interesting to show that these results also find an application in the
inference of the rates of $N$.

We begin by studying the probability mass function of the Markovian process
$N$ with (ordered) alternating parameters $\lambda_{1},\lambda_{2}>0$. By
means of formulas (3.7), (3.8) and Remark 3.3 of Cinque (2022) it is easy to
derive that, for $k\in\mathbb{N}_{0}$,

$P\\{N(t)=2k\\}=(\lambda_{1}t)^{k}(\lambda_{2}t)^{k}e^{-\lambda_{1}t}\,E^{k}_{1,2k+1}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)},$
(2.1)
$P\\{N(t)=2k+1\\}=(\lambda_{1}t)^{k+1}(\lambda_{2}t)^{k}e^{-\lambda_{1}t}\,E^{k+1}_{1,2k+2}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)},$
(2.2)

where
$E_{\nu,\delta}^{\gamma}(x)=\sum_{j=0}^{\infty}\frac{\Gamma(\gamma+j)}{\Gamma(\gamma)\,j!}\frac{x^{j}}{\Gamma(\nu
j+\delta)}$, with
$\nu,\gamma,\delta\in\mathbb{C},Re(\nu),Re(\gamma),Re(\delta)>0$ and
$x\in\mathbb{R}$, is the generalized Mittag-Leffler function. It can be proved
also that, for $0<t_{l}<t_{m}<t$ with $n\geq m>l\in\mathbb{N}$,

$\displaystyle P\\{T_{l}\in\mathop{}\\!\mathrm{d}t_{l},\ T_{m}\in$
$\displaystyle\mathop{}\\!\mathrm{d}t_{m}\ |\
N(t)=n\\}=\frac{t_{l}^{l-1}(t_{m}-t_{l})^{m-l-1}(t-t_{m})^{n-m}}{t^{n}}\>E^{\left\lfloor\frac{l}{2}\right\rfloor}_{1,l}\Bigl{(}t_{l}(\lambda_{1}-\lambda_{2})\Bigr{)}$
$\displaystyle\times\,\frac{E^{\left\lfloor\frac{m}{2}\right\rfloor-\left\lfloor\frac{l}{2}\right\rfloor}_{1,m-l}\Bigl{(}(t_{m}-t_{l})(\lambda_{1}-\lambda_{2})\Bigr{)}\,E^{\left\lfloor\frac{n+1}{2}\right\rfloor-\left\lfloor\frac{m}{2}\right\rfloor}_{1,n+1-m}\Bigl{(}(t-t_{m})(\lambda_{1}-\lambda_{2})\Bigr{)}}{E^{\left\lfloor\frac{n+1}{2}\right\rfloor}_{1,n+1}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}}\mathop{}\\!\mathrm{d}t_{l}\mathop{}\\!\mathrm{d}t_{m}.$
(2.3)

######  Remark 2.1.

Let $V(0)=v_{0},\ v_{0}\in\\{a_{1},a_{2}\\}$, and $v_{1}$ be the other
possible speed. By conditioning on the initial velocity $V(0)=v_{0}$ and the
number of changes of direction $N(t)=n\in\mathbb{N}$, in view of (1.2), we
have

$\displaystyle n=2k+1\implies\mathcal{T}(t)$
$\displaystyle=v_{0}T_{1}+v_{1}\bigl{(}T_{2}-T_{1}\bigr{)}+\dots+v_{0}\bigl{(}T_{2k+1}-T_{2k}\bigr{)}+v_{1}\bigl{(}t-T_{2k+1}\bigr{)}$
$\displaystyle=(v_{0}-v_{1})\bigl{(}T_{1}-T_{2}+T_{3}-\dots-
T_{2k}+T_{2k+1}\bigr{)}+v_{1}t$ (2.4)

and similarly in the even case
$n=2k\implies\mathcal{T}(t)=(v_{0}-v_{1})\bigl{(}T_{1}-T_{2}+T_{3}-\dots+T_{2k-1}-T_{2k}\bigr{)}+v_{0}t$.
Thus, conditionally on the number of switches in $(0,t)$ and the starting
speed, the telegraph process at time $t$ is an affine transformation of the
“alternating sums and differences” of the arrival times of $N(t)$.
Let $N(t)\geq n\in\mathbb{N}$, we define
$S_{n}(t):=\sum_{i=1}^{n}T_{i}(-1)^{i-1}$. Note that, if $N(t)=2k+1$, then
$S_{2k+1}(t)=T_{1}-T_{2}+T_{3}-\dots-
T_{2k}+T_{2k+1}=T_{1}+(T_{3}-T_{2})+\dots+(T_{2k+1}-T_{2k})$ represents the
time that the telegraph process spends with speed $V(0)$ and
$S_{2k+1}(t)\in(0,t)\ a.s.$. On the other hand, if $N(t)=2k$, $S_{2k}(t)$
represents the time that the particle spends with speed $V(0)$ minus the total
time $t$, then $S_{2k}(t)\in(-t,0)\ a.s.$ $\diamond$

Henceforth, we use the following notation. Let $(\Omega,\mathcal{F},P)$ be a
probability space and $X,Y:\Omega\longmapsto\mathbb{R}$ be absolutely
continuous random variables with joint probability density $f_{X,Y}$, for
$a,b,c,d,x,y\in\mathbb{R},a,c\not=0$, we write
$P\\{X\in(\mathop{}\\!\mathrm{d}x-b)/a,Y\in(\mathop{}\\!\mathrm{d}y-d)/c\\}:=f_{X,Y}\bigl{(}(x-b)/a,(y-d)/c\bigr{)}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y/|a\,c|=P\\{aX+b\in\mathop{}\\!\mathrm{d}x,cY+d\in\mathop{}\\!\mathrm{d}y\\}$.

######  Theorem 2.1.

Let $\\{N(t)\\}_{t\geq 0}$ be a Poisson process with ordered rates
$(\lambda_{1},\lambda_{2})$ and $S_{n}(t)$ defined as above. For
$k\in\mathbb{N},\ s\in(-t,0)$,

$P\\{S_{2k}(t)\in\mathop{}\\!\mathrm{d}s\ |\
N(t)=2k\\}=\frac{e^{-(\lambda_{1}-\lambda_{2})s}\>(-s)^{k-1}(t+s)^{k}}{E^{k}_{1,2k+1}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}\,k!(k-1)!\,t^{2k}}\mathop{}\\!\mathrm{d}s$
(2.5)

and, for $k\in\mathbb{N}_{0},\ s\in(0,t)$,

$P\\{S_{2k+1}(t)\in\mathop{}\\!\mathrm{d}s\ |\
N(t)=2k+1\\}=\frac{e^{(\lambda_{1}-\lambda_{2})(t-s)}\>s^{k}(t-s)^{k}}{E^{k+1}_{1,2k+2}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}\,k!^{2}\,t^{2k+1}}\mathop{}\\!\mathrm{d}s,$
(2.6)

where $E$ is the generalized Mittag-Leffler function.

###### Proof.

We prove (2.5) by induction. We begin by showing that the following equation
is true for $k\in\mathbb{N},\,t>0$.

$\displaystyle P\\{S_{2k}(t)\in\mathop{}\\!\mathrm{d}s\ |\ N(t)=2k\\}$ (2.7)
$\displaystyle=\frac{e^{-(\lambda_{1}-\lambda_{2})s}}{E^{k}_{1,2k+1}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}t^{2k}}\mathop{}\\!\mathrm{d}s\int_{0}^{t+s}\mathop{}\\!\mathrm{d}s_{2k-1}\int_{s}^{0}\mathop{}\\!\mathrm{d}s_{2k-2}\int_{0}^{s_{2k-1}}\mathop{}\\!\mathrm{d}s_{2k-3}\int_{s_{2k-2}}^{0}\mathop{}\\!\mathrm{d}s_{2k-4}\
\cdot\ \cdot\
\cdot\int_{s_{4}}^{0}\mathop{}\\!\mathrm{d}s_{2}\int_{0}^{s_{3}}\mathop{}\\!\mathrm{d}s_{1}.$

Formula (2.7) holds for $k=1$, in fact, for $s\in(-t,0)$,

$P\\{T_{1}-T_{2}\in\mathop{}\\!\mathrm{d}s\ |\
N(t)=2\\}=\int_{0}^{t+s}P\\{T_{1}\in\mathop{}\\!\mathrm{d}t_{1},\ T_{2}\in
t_{1}-\mathop{}\\!\mathrm{d}s\ |\ N(t)=2\\}$

and the base case is proved by suitably using (2.3). Let us suppose that (2.7)
holds for $k-1$ and note that, if $2h<n\in\mathbb{N}$ and real
$0<-s<r<t,\;P\\{S_{2h}(t)\in\mathop{}\\!\mathrm{d}s|N(t)=n,T_{2h}=r\\}=P\\{S_{2h}(r)\in\mathop{}\\!\mathrm{d}s|N(r)=2h\\}$.
Now, for $s\in(-t,0)$,

$\displaystyle P\\{$ $\displaystyle S_{2k}(t)\in\mathop{}\\!\mathrm{d}s\ |\
N(t)=2k\\}=\int_{0}^{t}P\\{S_{2k}(t)\in\mathop{}\\!\mathrm{d}s,\
S_{2k-1}(t)\in\mathop{}\\!\mathrm{d}s_{2k-1}\ |\ N(t)=2k\\}$
$\displaystyle=\int_{0}^{t+s}P\\{-T_{2k}\in\mathop{}\\!\mathrm{d}s-s_{2k-1},\
S_{2k-1}(t)\in\mathop{}\\!\mathrm{d}s_{2k-1}\ |\ N(t)=2k\\}$ (2.8)
$\displaystyle=\int_{0}^{t+s}\int_{s}^{0}P\\{T_{2k}\in
s_{2k-1}-\mathop{}\\!\mathrm{d}s,T_{2k-1}\in\mathop{}\\!\mathrm{d}s_{2k-1}-s_{2k-2},\
S_{2k-2}(t)\in\mathop{}\\!\mathrm{d}s_{2k-2}\ |\ N(t)=2k\\}$ (2.9)
$\displaystyle=\int_{0}^{t+s}\int_{s}^{0}P\\{T_{2k-1}\in\mathop{}\\!\mathrm{d}s_{2k-1}-s_{2k-2},\
T_{2k}\in s_{2k-1}-\mathop{}\\!\mathrm{d}s\ |\ N(t)=2k\\}$
$\displaystyle\hskip 56.9055pt\times
P\\{S_{2k-2}(s_{2k-1}-s_{2k-2})\in\mathop{}\\!\mathrm{d}s_{2k-2}\ |\
N(s_{2k-1}-s_{2k-2})=2k-2\\},$ (2.10)

where the integration sets of (2.8) and (2.9) follow by respectively
considering that $s_{2k-1}\in(0,t)$ and $s_{2k-1}-s\in(s_{2k-1},t)$ and that
$s_{2k-2}\in(-t,0)$ and $s_{2k-1}-s_{2k-2}\in(-s_{2k-2},\,s_{2k-1}-s)$. By
suitably plugging the induction hypothesis (2.7) (with $k-1$) and (2.3) into
(2.10), (2.7) shows up.

Finally, thanks to Fubini’s theorem, we can write (2.7) by separating the
integrals with odd and even indexed variables

$\displaystyle P\\{S_{2k}(t)\in\mathop{}\\!\mathrm{d}s\ |\ N(t)=2k\\}$
$\displaystyle=\frac{e^{-(\lambda_{1}-\lambda_{2})s}}{E^{k}_{1,2k+1}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}t^{2k}}\mathop{}\\!\mathrm{d}s\int_{0}^{t+s}\mathop{}\\!\mathrm{d}s_{2k-1}\int_{0}^{s_{2k-1}}\mathop{}\\!\mathrm{d}s_{2k-3}\cdots\int_{0}^{s_{3}}\mathop{}\\!\mathrm{d}s_{1}\,\int_{s}^{0}\mathop{}\\!\mathrm{d}s_{2k-2}\int_{s_{2k-2}}^{0}\mathop{}\\!\mathrm{d}s_{2k-4}\cdots\int_{s_{4}}^{0}\mathop{}\\!\mathrm{d}s_{2}.$

We conclude the proof of probability (2.5) by using the Cauchy integral
formula, i.e.
$\int^{x}_{0}\mathop{}\\!\mathrm{d}x_{1}\int^{x_{1}}_{0}\mathop{}\\!\mathrm{d}x_{2}\int^{x_{2}}_{0}\cdot\cdot\cdot\int^{x_{n-1}}_{0}f(x_{n})\mathop{}\\!\mathrm{d}x_{n}=\int_{0}^{x}f(x_{n})(x-x_{n})^{n-1}\mathop{}\\!\mathrm{d}x_{n}/\Gamma(n)$
with $f$ any integrable (real) function, $x\in\mathbb{R}$ and
$n\in\mathbb{N}$.

The proof of (2.6) works in the same way, but by means of the following
induction hypothesis, for $k\in\mathbb{N}_{0}$ and $s\in(0,t)$,

$\displaystyle P\\{S_{2k+1}(t)\in\mathop{}\\!\mathrm{d}s\ |\ N(t)=2k+1\\}$
$\displaystyle=\frac{e^{(\lambda_{1}-\lambda_{2})(t-s)}}{E^{k+1}_{1,2k+2}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}t^{2k+1}}\mathop{}\\!\mathrm{d}s\int^{0}_{s-t}\mathop{}\\!\mathrm{d}s_{2k}\int_{0}^{s}\mathop{}\\!\mathrm{d}s_{2k-1}\int^{0}_{s_{2k}}\mathop{}\\!\mathrm{d}s_{2k-2}\int^{s_{2k-1}}_{0}\mathop{}\\!\mathrm{d}s_{2k-3}\cdots\int_{s_{4}}^{0}\mathop{}\\!\mathrm{d}s_{2}\int_{0}^{s_{3}}\mathop{}\\!\mathrm{d}s_{1}.$

∎

######  Remark 2.2.

Let $n\in\mathbb{N}$ and $\mathbb{E}_{n}[\cdot]$ be the expected value with
respect to the probability measure $P\\{\cdot\,|\,N(t)=n\\}$. Put
$S_{n}=S_{n}(t)$. The $m$-th moment is, with $m>-k$ if $n=2k$ and $m>-k-1$ if
$n=2k+1$,

$\mathbb{E}_{2k}\bigl{[}S_{2k}^{m}\bigr{]}=(-t)^{m}\frac{\Gamma(k+m)}{(k-1)!}\frac{E^{k+m}_{1,2k+1+m}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}}{E^{k}_{1,2k+1}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}},$
$\mathbb{E}_{2k+1}\bigl{[}S_{2k+1}^{m}\bigr{]}=t^{m}\,\frac{\Gamma(k+m+1)}{k!}\frac{E^{k+1}_{1,2k+2+m}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}}{E^{k+1}_{1,2k+2}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}}.$

Clearly, for $m=0$ these expected values are both equal to $1$. $\diamond$

######  Remark 2.3.

Theorem 2.1 is useful to infer the ordered rates $(\lambda_{1},\lambda_{2})$
of the Poisson process $N$, whose probability mass appears in (2.1) and (2.2).
In fact, by observing the events recorded up to time $t>0$, the maximum
likelihood estimators are (put $S(t)=S_{N(t)}(t)$)
$\hat{\lambda}_{1}=\mathds{1}\bigl{(}N(t)\
\text{even}\bigr{)}\frac{N(t)/2}{t+S(t)}\,+\,\mathds{1}\bigl{(}N(t)\
\text{odd}\bigr{)}\frac{N(t)+1}{2\,S(t)},\ \
\hat{\lambda}_{2}=\mathds{1}\bigl{(}N(t)\
\text{even}\bigr{)}\frac{N(t)/2}{-S(t)}\,+\,\mathds{1}\bigl{(}N(t)\
\text{odd}\bigr{)}\frac{N(t)-1}{2(t-S(t))}.$

######  Theorem 2.2.

Let $\\{\mathcal{T}(t)\\}_{t\geq 0}$ be a telegraph process. Let
$x\in(a_{2}t,a_{1}t)$. For $k\in\mathbb{N}$,

$P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x\ |\ N(t)=2k,\
V(0)=a_{1}\\}=\frac{e^{(\lambda_{1}-\lambda_{2})\frac{a_{1}t-x}{a_{1}-a_{2}}}}{E^{k}_{1,2k+1}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}}\frac{(a_{1}t-x)^{k-1}(x-a_{2}t)^{k}}{k!(k-1)!\bigl{[}(a_{1}-a_{2})t\bigr{]}^{2k}}\mathop{}\\!\mathrm{d}x,$
(2.11) $P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x\ |\ N(t)=2k,\
V(0)=a_{2}\\}=\frac{e^{(\lambda_{1}-\lambda_{2})\frac{a_{1}t-x}{a_{1}-a_{2}}}}{E^{k+1}_{1,2k+1}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}}\frac{\bigl{(}a_{1}t-x\bigr{)}^{k}\bigl{(}x-a_{2}t\bigr{)}^{k-1}}{k!(k-1)!\,\bigl{[}(a_{1}-a_{2})t\bigr{]}^{2k}}\mathop{}\\!\mathrm{d}x,$
(2.12)

and for $k\in\mathbb{N}_{0},\ v_{0}\in\\{a_{1},a_{2}\\}$,

$P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x\ |\ N(t)=2k+1,\
V(0)=v_{0}\\}=\frac{e^{(\lambda_{1}-\lambda_{2})\frac{a_{1}t-x}{a_{1}-a_{2}}}}{E^{k+1}_{1,2k+2}\Bigl{(}t(\lambda_{1}-\lambda_{2})\Bigr{)}}\frac{(a_{1}t-x)^{k}(x-a_{2}t)^{k}}{k!^{2}\bigl{[}(a_{1}-a_{2})t\bigr{]}^{2k+1}}\mathop{}\\!\mathrm{d}x.$
(2.13)

The distributions in Theorem 2.2 can be found in Lopez and Ratanov (2014)
(where the confluent hypergeometric Kummer function is involved).

###### Proof.

The proof follows by considering Remark 2.1, Theorem 2.1 and that $N$ has
parameter $(\lambda_{1},\lambda_{2})$ if $V(0)=a_{1}$ and
$(\lambda_{2},\lambda_{1})$ if $V(0)=a_{2}$. Also note that
$E_{1,\gamma_{1}+\gamma_{2}}^{\gamma_{1}}(y-x)=e^{y-x}E_{1,\gamma_{1}+\gamma_{2}}^{\gamma_{2}}(x-y)$,
with $\gamma_{1},\gamma_{2}\in\mathbb{C},\ Re(\gamma_{1}),Re(\gamma_{2})>0$
and $x,y\in\mathbb{R}$ (see Remark 3.3 of Cinque (2022)). ∎

Note that if $N(t)=2k+1$, the motion performs $k+1$ displacements with each
velocity and, thanks to the Markovianity of $N$, the initial velocity is not
relevant. For instance, if we consider $N$ with Mittag-Leffler distributed
waiting times, the process loses this property, see Theorem 3.4 of Cinque
(2022).

From Theorem 2.2, with (2.1) and (2.2) at hand, we obtain the well-known
absolutely continuous component of the telegraph process. Let
$v_{0}\in\\{a_{1},a_{2}\\}$ and $v_{1}$ the other possible speed. For
$x\in(a_{2}t,a_{1}t)$

$\displaystyle
P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x\,|\,V(0)=v_{0}\\}=\frac{e^{{-\frac{\lambda_{1}(x-a_{2}t)+\lambda_{2}(a_{1}t-x)}{a_{1}-a_{2}}}}}{a_{1}-a_{2}}\mathop{}\\!\mathrm{d}x$
(2.14)
$\displaystyle\times\Biggl{[}\lambda_{h}\,I_{0}\Bigl{(}\frac{2\sqrt{\lambda_{1}\lambda_{2}}}{a_{1}-a_{2}}\sqrt{(a_{1}t-x)(x-a_{2}t)}\Bigr{)}+\sqrt{\lambda_{1}\lambda_{2}}\sqrt{\frac{|v_{1}t-x|}{|v_{0}t-x|}}\,I_{1}\Bigl{(}\frac{2\sqrt{\lambda_{1}\lambda_{2}}}{a_{1}-a_{2}}\sqrt{(a_{1}t-x)(x-a_{2}t)}\Bigr{)}\Biggr{]},$

where $h=\Big{\\{}\begin{array}[]{l l}1&\text{if}\ v_{0}=a_{1}\\\ 2&\text{if}\
v_{0}=a_{2}\end{array}$ and
$I_{\nu}(x)=\sum_{k=0}^{\infty}\frac{(x/2)^{2k+\nu}}{k!\Gamma(k+1+\nu)}$ is
the modified Bessel function of order $\nu\in\mathbb{R}$.

##  3 Probability law of the telegraph process conditioned on the position at
previous time

The counting process $N$ has independent waiting times, thus, every time a
change of direction occurs, the motion $\mathcal{T}$ starts again with the
same characteristics, independently on the previous displacements (except for
the order of the rates, given by the current velocity). As above, we denote
with $T_{k}<t$ the $k$-th arrival time of the process $N(t)=n\geq
k\in\mathbb{N}$ and with $V(T_{k})=v_{k}$ the speed at time $T_{k}$ after the
change of direction. It is clear that for $x\in[a_{2}T_{k},a_{1}T_{k}]$ and
$y\in\bigr{(}x+a_{2}(t-T_{k}),x+a_{1}(t-T_{k})\bigl{)}$,

$\displaystyle P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}y\,$
$\displaystyle|\,\mathcal{T}(T_{k})=x,\,N(T_{k})=k,\,N(t)=n,\,V(0)=v_{0}\\}$
(3.1)
$\displaystyle=P\\{\mathcal{T}(t-T_{k})\in\mathop{}\\!\mathrm{d}y-x\,|\,N(t-T_{k})=n-k,\,V(0)=v_{k}\\}.$

Furthermore, when $N$ is Markovian, the couple
$\big{\\{}\bigl{(}\mathcal{T}(t),V(t)\bigr{)}\big{\\}}_{t\geq 0}$ is a Markov
process. Now, by considering that the event $\\{N(t)=k,V(0)=v_{0}\\}$ implies
that $V(t)=v_{k}$ is known, we immediately obtain the following result (see
Appendix A for an alternative proof of Theorem 3.1).

######  Theorem 3.1.

Let $\\{\mathcal{T}(t)\\}_{t\geq 0}$ be a telegraph process. Let $0<s<t,\
x\in[a_{2}s,a_{1}s]$ and $v_{0}\in\\{a_{1},a_{2}\\}$. For
$k\in\mathbb{N}_{0},\ (y-x)\in\bigr{(}a_{2}(t-s),a_{1}(t-s)\bigl{)}$,

$P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}y\ |\
\mathcal{T}(s)=x,N(s)=k,V(0)=v_{0}\\}=P\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\
|\ V(0)=v_{k}\\}$ (3.2)

and

$P\\{\mathcal{T}(t)=x+v_{k}(t-s)\ |\
\mathcal{T}(s)=x,N(s)=k,V(0)=v_{0}\\}=P\\{N(t-s)=0\ |\ V(0)=v_{k}\\},$ (3.3)

with $v_{k}=V(T_{k})$ being the speed after the $k$-th change of direction.

We consider the asymmetric telegraph process
($\lambda_{1}=\lambda_{2}=\lambda$). The interested reader can obtain the
results for the telegraph process with two different rates by following the
same steps we show below.

######  Remark 3.1.

We recall that if $\\{\mathcal{T}(t)\\}_{t\geq 0}$ is an asymmetric telegraph
process, then, for $n\in\mathbb{N},\ x\in(a_{2}t,a_{1}t),\
v_{0}\in\\{a_{1},a_{2}\\}$ and $v_{1}$ being the other possible velocity,

$P\\{V(0)=v_{0}\ |\ \mathcal{T}(t)=x,N(t)=n\\}=\begin{cases}\begin{array}[]{l
l}\frac{|v_{1}t-x|}{(a_{1}-a_{2})t}&\text{if}\ n\ \text{even,}\\\
\frac{1}{2}&\text{if}\ n\ \text{odd.}\end{array}\end{cases}$ (3.4)

Note that the odd case in (3.4) holds true also for the telegraph process with
two different rates.

We point out that for $k\in\mathbb{N},\ P\\{V(0)=a_{1}\ |\
\mathcal{T}(t)=x,N(t)=2k\\}\geq\frac{1}{2}\iff x\geq(a_{1}+a_{2})t/2$.
Clearly, $V(0)=V(t)$ because an even number of changes of direction occurred
in $[0,t]$ and thus, at time $t$, it is more likely that the motion has
velocity $a_{1}$ if and only if $|x-a_{1}t|<|x-a_{2}t|$. Note that in the case
of the symmetric telegraph process ($a_{1}=-a_{2}>0$) $P\\{V(0)=v_{0}\ |\
\mathcal{T}(t)=x,N(t)=2k\\}>\frac{1}{2}\iff sign(v_{0})=sign(x)$. $\diamond$

By means of simple probability elaborations, the previous results permit us to
obtain Theorem 3.2 and Corollary 3.1. These provide the explicit formulas for
the conditional distributions of the telegraph process as the sum of the
“Markov” term, that is the probability that would appear if Markovianity held,
and an additional term.

######  Theorem 3.2.

Let $\\{\mathcal{T}(t)\\}_{t\geq 0}$ be an asymmetric telegraph process. Let
$0<s<t,\ x\in(a_{2}s,a_{1}s),\ y\in\bigl{(}x+a_{2}(t-s),x+a_{1}(t-s)\bigr{)},\
v_{0}\in\\{a_{1},a_{2}\\}$ and $v_{1}$ be the other possible speed. For
$k\in\mathbb{N}_{0}$,

$P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}y\ |\
\mathcal{T}(s)=x,N(s)=2k+1\\}=P\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\\}$
(3.5)

and

$P\\{\mathcal{T}(t)=x+v_{0}(t-s)\ |\
\mathcal{T}(s)=x,N(s)=2k+1\\}=P\\{\mathcal{T}(t-s)=v_{0}(t-s)\\}.$ (3.6)

For $k\in\mathbb{N}$,

$P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}y\ |\
\mathcal{T}(s)=x,N(s)=2k\\}=P\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\\}+g(s,t-s;x,y-x)\mathop{}\\!\mathrm{d}y,$
(3.7)

where

$\displaystyle g(s,$ $\displaystyle
t-s;x,y-x)=\frac{I_{1}\Bigl{(}\frac{2\lambda}{a_{1}-a_{2}}\sqrt{\bigl{[}a_{1}(t-s)-(y-x)\bigr{]}\bigl{[}(y-x)-a_{2}(t-s)\bigr{]}}\Bigr{)}}{\sqrt{\bigl{[}a_{1}(t-s)-(y-x)\bigr{]}\bigl{[}(y-x)-a_{2}(t-s)\bigr{]}}}$
(3.8) $\displaystyle\ \ \ \ \times\frac{\lambda
e^{-\lambda(t-s)}}{2(a_{1}-a_{2})^{2}s}\Bigl{[}4x(y-x)+(a_{1}+a_{2})\bigl{[}(a_{1}+a_{2})s(t-s)-2s(y-x)-2(t-s)x\bigr{]}\Bigr{]}$
$\displaystyle=e^{-\lambda(t-s)}\,\frac{(a_{1}+a_{2})s-2x}{2(a_{1}-a_{2})s}\,\frac{\partial}{\partial
y}I_{0}\Bigl{(}\frac{2\lambda}{a_{1}-a_{2}}\sqrt{\bigl{[}a_{1}(t-s)-(y-x)\bigr{]}\bigl{[}(y-x)-a_{2}(t-s)\bigr{]}}\Bigr{)}$
(3.9)

and

$P\\{\mathcal{T}(t)=x+v_{0}(t-s)\ |\
\mathcal{T}(s)=x,N(s)=2k\\}=\frac{|v_{1}s-x|}{(a_{1}-a_{2})s}P\\{\mathcal{T}(t-s)=v_{0}(t-s)\\}.$
(3.10)

Note that all the results are independent of $k$. Moreover, (3.5) and (3.6)
hold true for the telegraph process with two different rates.

###### Proof.

The proof follows from Theorem 3.1 and Remark 3.1. To prove (3.7), put
$P_{i}\\{\,\cdot\,\\}=P\\{\,\cdot\,|\,V(0)=a_{i}\\},\ i=1,2$, then

$\displaystyle P$ $\displaystyle\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}y\
|\
\mathcal{T}(s)=x,N(s)=2k\\}=\frac{(x-a_{2}s)P_{1}\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\\}+(a_{1}s-x)P_{2}\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\\}}{(a_{1}-a_{2})s}$
$\displaystyle=2P\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\\}-\frac{(a_{1}s-x)P_{1}\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\\}+(x-a_{2}s)P_{2}\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\\}}{(a_{1}-a_{2})s}$

and by suitably applying formula (2.14), some calculation yield the claimed
result. ∎

By taking into account the asymptotic behavior of the Bessel function, for
$\nu=0,1$, $I_{\nu}(x)\sim e^{x}/\sqrt{2\pi x}$, with
$x\longrightarrow\infty$, we can prove that (3.8) converges to $0$ under the
generalized Kac’s conditions (see above, after equation (1.3)). Furthermore,
thanks to expression (3.9) we readily obtain that (3.7) satisfies the
differential equation (1.3) with time variable $t$ and space variable $y$ and
that its integral on $\bigl{(}x+a_{2}(t-s),x+a_{1}(t-s)\bigr{)}$ is equal to
$1-e^{-\lambda(t-s)}$.

######  Remark 3.2 (Symmetric telegraph process).

Let us assume the velocities $a_{1}=-a_{2}=c>0$ and the rate
$\lambda_{1}=\lambda_{2}=\lambda>0$. Function $g$ in (3.8) substantially
simplifies. We can write $g=x(y-x)f$ where

$\displaystyle x(y-x)f(s,t-s;y-x)=\frac{x(y-x)\,\lambda
e^{-\lambda(t-s)}}{2c^{2}s\sqrt{c^{2}(t-s)^{2}-(y-x)^{2}}}\,I_{1}\Bigl{(}\frac{\lambda}{c}\sqrt{c^{2}(t-s)^{2}-(y-x)^{2}}\Bigr{)}$
(3.11)

$f$ is positive for $(y-x)\in\bigl{(}c(t-s),c(t-s)\bigr{)}$. Then, (3.11) is
positive if $sign(y)=sign(x)$ and $|y|>|x|$, so if $y$ is further than $x$
from the origin. This happens because the probability mass of
$V(0)=V(s)=sign(x)c$ is greater than the probability of the opposite velocity
at time $s$ (see Remark 3.1). $\diamond$

######  Corollary 3.1.

Let $\\{\mathcal{T}(t)\\}_{t\geq 0}$ be an asymmetric telegraph process. Let
$0<s<t,\ x\in(a_{2}s,a_{1}s)$. For
$y\in\bigl{(}x+a_{2}(t-s),x+a_{1}(t-s)\bigr{)}$,

$P\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}y\ |\
\mathcal{T}(s)=x\\}=P\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\\}\,+\,g(s,t-s;x,y-x)\mathop{}\\!\mathrm{d}y\,P\\{N(s)\
\text{even}\ |\ \mathcal{T}(s)=x\\},$ (3.12)

where $g$ is defined in (3.8). For $v_{0}\in\\{a_{1},a_{2}\\}$ and $v_{1}$
being the other possible velocity,

$P\\{\mathcal{T}(t)=x+v_{0}(t-s)\ |\
\mathcal{T}(s)=x\\}=\frac{e^{-\lambda(t-s)}}{2}\Bigl{(}1+\frac{|v_{1}s-x|}{(a_{1}-a_{2})s}\,P\\{N(s)\
\text{even}\ |\ \mathcal{T}(s)=x\\}\Bigr{)}.$ (3.13)

###### Proof.

The corollary is an easy consequence of Theorem 3.2. ∎

We recall that for $0<s<t,\ x\in(a_{2}s,a_{1}s)$, by setting
$A(s,x)=\sqrt{(a_{1}s-x)(x-a_{2}s)}$, we have

$P\\{N(s)\ \text{even}\ |\
\mathcal{T}(s)=x\\}=\frac{(a_{1}-a_{2})s\,I_{1}\bigl{(}\frac{2\lambda}{a_{1}-a_{2}}A(s,x)\bigr{)}}{2A(s,x)\,I_{0}\bigl{(}\frac{2\lambda}{a_{1}-a_{2}}A(s,x)\bigr{)}+(a_{1}-a_{2})s\,I_{1}\bigl{(}\frac{2\lambda}{a_{1}-a_{2}}A(s,x)\bigr{)}}$
(3.14)

and (3.14) converges to $1/2$ under the generalized Kac’s conditions. Thus,
the asymmetric telegraph process (with $a_{1}>0>a_{2}$) at time $t>0$,
conditionally on $\mathcal{T}(s)=x$, converges in distribution to the process
$B(t-s)+x$ with $\\{B(t)\\}_{t\geq 0}$ being a standard Brownian motion with
drift $\mu\in\mathbb{R}$ ($a_{1}+a_{2}\longrightarrow-2\mu$). Clearly, (3.12)
satisfies the differential equation (1.3) with time variable $t$ and space
variable $y$.

The interested reader can now obtain the joint distribution of the telegraph
process at two distinct times as well as the distribution of the telegraph
bridge.

##  4 Conditional probability of the symmetric telegraph process with its
maximum and its minimum

Let $m(t):=\min_{0\leq s\leq t}\mathcal{T}(s)$ and $M(t):=\max_{0\leq s\leq
t}\mathcal{T}(s)$ be respectively the minimum and the maximum of the symmetric
($\lambda_{1}=\lambda_{2}=\lambda>0,a_{1}=-a_{2}=c>0$) telegraph process
$\mathcal{T}$, in the time interval $[0,t]$. We denote with
$P^{\pm}_{n}(\cdot)=P\\{\ \cdot\ |\ V(0)=\pm c,\ N(t)=n\\}$ the probability
measure conditioned on the starting velocity and the number of switches up to
time $t\geq 0$.

The conditional and unconditional distributions of the maximum/minimum of the
(symmetric) telegraph process are well known, see Foong and Kanno (1994),
Cinque and Orsingher (2020, 2021) (for the first passage times see also De
Bruyne et al. (2021), Ratanov (2021)). The joint distributions with the
position of the particle at the ending time are known as well, see Cinque
(2020). For the sake of clarity, we observe that, for any integer $n\geq 0$,
real $x$ and $\beta$, thanks to the symmetry of the motion with constant rate
and velocities $\pm c$, we have

$P^{\pm}_{n}\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x,\
m(t)<-\beta\\}=P^{\mp}_{n}\\{\mathcal{T}(t)\in-\mathop{}\\!\mathrm{d}x,\
M(t)>\beta\\}$ (4.1)

Let $\alpha\in[0,ct)$ and $\beta\in[0,ct)$. We focus on the probability that
the telegraph process moves both below level $-\alpha$ and above $\beta$ and
it ends in $x\in[-\alpha,\beta]$ in the time interval $[0,t]$. Thus, we
consider the following distribution, for natural $n\geq 2$ (if $n=0,1$ it is
equal to zero),

$\displaystyle P_{n}^{+}\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x,\ m(t)<$
$\displaystyle-\alpha,\
M(t)>\beta\\}=P^{-}_{n}\\{\mathcal{T}(t)\in-\mathop{}\\!\mathrm{d}x,\
m(t)<-\beta,\ M(t)>\alpha\\}$ (4.2)
$\displaystyle=P_{n}^{+}\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x,\
m(t)<-\alpha,\ M(t)>\beta,\ F_{-\alpha}>F_{\beta}\\}$ (4.3) $\displaystyle\ \
\ \ +\ P_{n}^{+}\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x,\ m(t)<-\alpha,\
M(t)>\beta,\ F_{-\alpha}<F_{\beta}\\}$ (4.4)

with $F_{y}$ being the first passage time of the telegraph process across
level $y\in\mathbb{R}$. We say that the above probabilities are trivial if
they reduce to known distributions (for instance if
$x\in(-ct,-\alpha)\cup(\beta,ct)$).

######  Proposition 4.1.

Let $\\{\mathcal{T}(t)\\}_{t\geq 0}$ be a symmetric telegraph process.
Distribution (4.3) is non-trivial and not null for
$(\beta,-\alpha,x)\in\mathcal{S}_{M}$, where

$\begin{array}[]{l}\mathcal{S}_{M}=\Big{\\{}\;(\beta,-\alpha,x)\,:\,\Bigl{(}\;\beta\in\bigl{[}0,\frac{ct}{3}\bigr{)},\,-\alpha\in\bigl{(}\frac{3\beta-
ct}{2},0\bigr{]},\,x\in\bigl{[}-\alpha,\beta\bigr{]}\;\Bigr{)}\\\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ or\
\Bigl{(}\;\beta\in\bigl{[}0,\frac{ct}{3}\bigr{)},\,-\alpha\in\bigl{(}2\beta-
ct,\frac{3\beta-
ct}{2}\bigr{]},\,x\in\bigl{[}-\alpha,ct-2\alpha-2\beta\bigr{]}\;\Bigr{)}\\\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ or\
\Bigl{(}\;\beta\in\bigl{[}\frac{ct}{3},\frac{ct}{2}\bigr{)},\,-\alpha\in\bigl{(}2\beta-
ct,0\bigr{]},\,x\in\bigl{[}-\alpha,ct-2\alpha-2\beta\bigr{]}\;\Bigr{)}\
\Big{\\}}\end{array}$ (4.5)

and natural $n\geq 2$. Distribution (4.4) is non-trivial and not null for
$(\beta,-\alpha,x)\in\mathcal{S}_{m}$, where

$\begin{array}[]{l}\mathcal{S}_{m}=\Big{\\{}\;(\beta,-\alpha,x)\,:\,\Bigl{(}\;\beta\in\big{[}0,\frac{ct}{2}\bigr{)},\,-\alpha\in\bigl{(}\frac{2\beta-
ct}{3},0\bigr{]},\,x\in\bigl{[}-\alpha,\beta\bigr{]}\;\Bigr{)}\\\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ or\
\Bigl{(}\;\beta\in\bigl{[}0,\frac{ct}{2}\bigr{)},\,-\alpha\in\bigl{(}\frac{\beta-
ct}{2},\frac{2\beta-ct}{3}\bigr{]},\,x\in\bigl{[}2\alpha+2\beta-
ct,\beta\bigr{]}\;\Bigr{)}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
or\
\Bigl{(}\;\beta\in\bigl{[}\frac{ct}{2},ct\bigr{)},\,-\alpha\in\bigl{(}\frac{\beta-
ct}{2},0\bigr{]},\,x\in\bigl{[}2\alpha+2\beta-ct,\beta\bigr{]}\;\Bigr{)}\
\Big{\\}}\end{array}$ (4.6)

and natural $n\geq 3$.

###### Proof.

Since $F_{\beta}<F_{-\alpha}$, we need $\beta<\frac{ct}{2}$ in order to pass
$\beta$ and then move lower than $-\alpha<0$. Thus $\beta\in[0,\frac{ct}{2})$.
Now, we want $\alpha\in[0,ct)$ such that
$c(t-\frac{\beta}{c})>|\beta+\alpha|$, meaning that the motion has time to
reach $-\alpha$ after it crossed $\beta$. Thus $-\alpha\in(2\beta-ct,0]$. At
last, we require $x\in[-\alpha,\beta]$ such that
$c(t-\frac{\beta}{c}-\frac{\beta+\alpha}{c})\geq x+\alpha$, therefore
$x\in[-\alpha,\min\\{\beta,ct-2\alpha-2\beta\\}]$. Finally,
$\min\\{\beta,ct-2\alpha-2\beta\\}=\beta$ if $-\alpha>\frac{3\beta-ct}{2}$ and
$2\beta-ct<\frac{3\beta-ct}{2}<0$ if $\beta<\frac{ct}{3}$, then we have (4.5).
By means of similar arguments we obtain (4.6). ∎

We now show a general recurrent formula for probability (4.3). We observe the
following:

  * ($i$)

if $\mathcal{T}$ moves beyond $\beta$ during the first displacement, it must
have time to both reach $-\alpha$ and to be at level $x$ at time $t$, then
$T_{1}\geq\beta/c$ and $c(t-T_{1})\geq(cT_{1}+\alpha)+(x+\alpha)$;

  * ($ii$)

if $T_{1}<\beta/c$, it is necessary that
$-\alpha<\mathcal{T}(T_{2})=2cT_{1}-cT_{2}$, because $F_{\beta}<F_{-\alpha}$,
and $c(t-T_{2})\geq\beta-\mathcal{T}(T_{2})+(\beta+\alpha)+(x+\alpha)$.
Therefore,
$T_{2}\leq\min\big{\\{}\frac{2cT_{1}+\alpha}{c},\frac{ct-2\alpha-2\beta-x}{2c}+T_{1}\big{\\}}=\frac{2cT_{1}+\alpha}{c}$
if $T_{1}\leq\frac{ct-4\alpha-2\beta-x}{2c}$.

With this at hand, by keeping in mind (3.1), thanks to a recurrence argument
on the displacements of the motion (see for instance the proof of Theorem 3.1
of Cinque and Orsingher (2020)), we can write the following relationship for
distribution (4.3). Let natural $n\geq 2$ and
$(\beta,-\alpha,x)\in\mathcal{S}_{M}$, then

$\displaystyle P_{n}^{+}\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x,\
m(t)<-\alpha,\ M(t)>\beta,\ F_{-\alpha}>F_{\beta}\\}$ (4.7)
$\displaystyle=\int_{\frac{\beta}{c}}^{\frac{ct-2\alpha-x}{2c}}P^{-}_{n-1}\\{\mathcal{T}(t-t_{1})\in\mathop{}\\!\mathrm{d}x-ct_{1},\
m(t-t_{1})<-\alpha-ct_{1}\\}P\\{T_{1}\in\mathop{}\\!\mathrm{d}t_{1}|N(t)=n\\}$
(4.8) $\displaystyle\ \
+\begin{cases}\int_{0}^{\frac{\beta}{c}}\mathop{}\\!\mathrm{d}t_{1}\int_{t_{1}}^{\frac{ct-2\alpha-2\beta-x}{2c}+t_{1}}p^{+}_{n-2}(t_{1},t_{2})\mathop{}\\!\mathrm{d}t_{2},\hfill{\text{if}\
\frac{ct-4\alpha-2\beta-x}{2c}\leq 0\leq\frac{\beta}{c}},\\\\[10.0pt]
\int_{0}^{\frac{ct-4\alpha-2\beta-x}{2c}}\mathop{}\\!\mathrm{d}t_{1}\int_{t_{1}}^{\frac{2ct_{1}+\alpha}{c}}p^{+}_{n-2}(t_{1},t_{2})\mathop{}\\!\mathrm{d}t_{2}+\int_{\frac{ct-4\alpha-2\beta-x}{2c}}^{\frac{\beta}{c}}\mathop{}\\!\mathrm{d}t_{1}\int_{t_{1}}^{\frac{ct-2\alpha-2\beta-x}{2c}+t_{1}}p^{+}_{n-2}(t_{1},t_{2})\mathop{}\\!\mathrm{d}t_{2},\\\
\hfill{\text{if}\ 0<\frac{ct-4\alpha-2\beta-x}{2c}\leq\frac{\beta}{c}},\\\
\int_{0}^{\frac{\beta}{c}}\mathop{}\\!\mathrm{d}t_{1}\int_{t_{1}}^{\frac{2ct_{1}+\alpha}{c}}p^{+}_{n-2}(t_{1},t_{2})\mathop{}\\!\mathrm{d}t_{2},\hfill{\text{if}\
0<\frac{\beta}{c}<\frac{ct-4\alpha-2\beta-x}{2c}},\end{cases}$ (4.9)

where

$\displaystyle p^{+}_{n-2}(t_{1},t_{2})$
$\displaystyle\mathop{}\\!\mathrm{d}t_{1}\mathop{}\\!\mathrm{d}t_{2}=P\\{T_{1}\in\mathop{}\\!\mathrm{d}t_{1},T_{2}\in\mathop{}\\!\mathrm{d}t_{2}\
|\
N(t)=n\\}\,P_{n-2}^{+}\\{\mathcal{T}(t-t_{2})\in\mathop{}\\!\mathrm{d}x-2ct_{1}+ct_{2},$
(4.10) $\displaystyle
m(t-t_{2})<-\alpha-2ct_{1}+ct_{2},M(t-t_{2})>\beta-2ct_{1}+ct_{2},\,F_{-\alpha-2ct_{1}+ct_{2}}>F_{\beta-2ct_{1}+ct_{2}}\\}.$

We point out that Proposition 4.1 and the recurrence formula (4.7) hold true
even if we consider a telegraph process with alternating rates
$\lambda_{1},\lambda_{2}>0$ (and also for a more general Markovian counting
process $N$).

Clearly if $n=2,3$ formula (4.7) reduces to term (4.8) only. By means of (4.1)
and Theorem 3.1 of Cinque (2020), it is easy to prove that (4.8) reads, with
natural $n\geq 2$,

$\begin{cases}\begin{array}[]{l
l}\displaystyle\frac{(2k)!}{(k-1)!k!}\frac{(ct+2\alpha+x)^{k-1}(ct-2\alpha-2\beta-x)^{k}}{(2ct)^{2k}}\mathop{}\\!\mathrm{d}x,&\text{if}\
n=2k,\\\\[8.0pt]
\displaystyle\frac{(2k+1)!}{(k-1)!(k+1)!}\frac{(ct+2\alpha+x)^{k-1}(ct-2\alpha-2\beta-x)^{k+1}}{(2ct)^{2k+1}}\mathop{}\\!\mathrm{d}x,&\text{if}\
n=2k+1.\end{array}\end{cases}$ (4.11)

######  Proposition 4.2.

Let $\\{\mathcal{T}(t)\\}_{t\geq 0}$ be a symmetric telegraph process. Let
natural $n\geq 2$. For $(\beta,-\alpha,x)\in\mathcal{S}_{M}$ and
$\frac{ct-4\alpha-2\beta-x}{2c}\leq 0\leq\frac{\beta}{c}$,

$\displaystyle P_{n}^{+}\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x,\
m(t)<-\alpha,\ M(t)>\beta,\
F_{-\alpha}>F_{\beta}\\}=P^{-}_{n}\\{\mathcal{T}(t)\in
2\beta-\mathop{}\\!\mathrm{d}x,\ M(t)>2\beta+\alpha\\}$ (4.12)
$\displaystyle=\begin{cases}\displaystyle P_{2k}^{-}\\{\mathcal{T}(t)\in
2\alpha+2\beta+\mathop{}\\!\mathrm{d}x\\}=\frac{(2k)!}{(k-1)!k!}\frac{(ct+2\alpha+2\beta+x)^{k-1}(ct-2\alpha-2\beta-x)^{k}}{(2ct)^{2k}}\mathop{}\\!\mathrm{d}x,&\\\
\hfill\text{if}\ n=2k,\\\\[8.0pt]
\displaystyle\frac{(2k+1)!}{(k-1)!(k+1)!}\frac{(ct+2\alpha+2\beta+x)^{k-1}(ct-2\alpha-2\beta-x)^{k+1}}{(2ct)^{2k+1}}\mathop{}\\!\mathrm{d}x,\hfill\text{if}\
n=2k+1.\end{cases}$

###### Proof.

Since $\frac{ct-4\alpha-2\beta-x}{2c}\leq 0\leq\frac{\beta}{c}$, the
proposition follows by summing up (4.11) and the first case of (4.9). The
explicit form of the latter is obtained by induction and by suitably applying
the results of Corollary 3.4 and Corollary 3.5 of Cinque (2020) (see also
Theorem 3.1 of Cinque and Orsingher (2020) for a similar induction method). ∎

Intuitively, relationship (4.12) holds true because, when
$\frac{ct-4\alpha-2\beta-x}{2c}\leq 0\leq\frac{\beta}{c}$, the motion can not
reach level $x=-\alpha$ before crossing the threshold $x=\beta$. Formula
(4.12) resembles the negative reflection principle for the telegraph process
and it can be graphically described in a similar way, see Figures 2 and 2 (and
see Section 4 of Cinque (2020) for all the details).

$\beta$$-\alpha$$t_{1}$A$t_{2}$BC$t_{3}$D$t$$x$E$\boldsymbol{s}$$\boldsymbol{\mathcal{T}(s)}$O
Figure 1: Sample path $\omega^{+}$ with $N(t)=8\ $ and
$\frac{ct-4\alpha-2\beta-x}{2c}\leq 0\leq\frac{\beta}{c}$.

$\beta$$-\alpha$$2\beta$$2\beta+\alpha$O’$t_{2}$BC’
$t_{3}$D’$t$$2\beta-x$E’$\boldsymbol{s}$$\boldsymbol{\mathcal{T}(s)}$A’ Figure
2: The negatively reflected sample of $\omega^{+}$.

At last, we can write probability (4.4) in terms of (4.3). Let natural $n\geq
3$ and $(\beta,-\alpha,x)\in\mathcal{S}_{m}$,

$\displaystyle P_{n}^{+}$
$\displaystyle\\{\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}x,\ m(t)<-\alpha,\
M(t)>\beta,\
F_{-\alpha}<F_{\beta}\\}=\int_{0}^{\min\\{\frac{\beta}{c},\frac{ct-2\alpha-2\beta+x}{2c}\\}}P\\{T_{1}\in\mathop{}\\!\mathrm{d}t_{1}\
|\ N(t)=n\\}$ (4.13) $\displaystyle\times
P_{n-1}^{+}\\{\mathcal{T}(t-t_{1})\in ct_{1}-\mathop{}\\!\mathrm{d}x,\
m(t-t_{1})<ct_{1}-\beta,\ M(t-t_{1})>\alpha+ct_{1},\
F_{\alpha+ct_{1}}<F_{ct_{1}-\beta}\\}.$

By keeping in mind Proposition 4.1, we obtain that

$\min\Big{\\{}\frac{\beta}{c},\;\frac{ct-2\alpha-2\beta+x}{2c}\Big{\\}}=\frac{\beta}{c}\,\mathds{1}_{\mathcal{R}}(\beta,-\alpha,x)\
+\
\frac{ct-2\alpha-2\beta+x}{2c}\,\mathds{1}_{\mathcal{S}_{m}\setminus\mathcal{R}}(\beta,-\alpha,x),$
(4.14)

with $\mathds{1}_{A}$ being the indicator function of a set $A$ and
$\mathcal{R}=\Big{\\{}(\beta,-\alpha,x)\,:\,\beta\in\bigl{[}0,\frac{ct}{3}\bigr{]},\,-\alpha\in\bigl{[}\frac{3\beta-
ct}{2},0\bigr{]},\,x\in\bigl{[}\max\\{-\alpha,\ 2\alpha+4\beta-
ct\\},\,\beta\bigr{]}\Big{\\}}$.

##  Appendix A Proof of Theorem 3.1

Probability (3.3) concerns the case where no changes of direction occur in the
time interval $(s,t)$, then it immediately follows from Markovianity of $N$.
For $0<w<t$, let $\mathcal{T}(t)=\mathcal{T}(w)+\mathcal{T}_{w}(t)$, where the
process $\\{\mathcal{T}_{w}(s)\\}_{s\geq w}$ describes the evolution of
$\mathcal{T}$ in the time interval $[w,s]$ with respect to the position
$\mathcal{T}(w)$, meaning that $\mathcal{T}_{w}(w)=0\ a.s.$. By bearing in
mind (3.1), if the motion changes direction at time $w$, then
$\mathcal{T}_{w}(t)\stackrel{{\scriptstyle d}}{{=}}\mathcal{T}(t-w)$. Thus,

$\displaystyle P\\{$ $\displaystyle\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}y\
|\ \mathcal{T}(s)=x,N(s)=k,N(t)=n,V(0)=v_{0}\\}$
$\displaystyle=P\\{\mathcal{T}(s)+v_{k}(T_{k+1}-s)+\mathcal{T}_{T_{k+1}}(t)\in\mathop{}\\!\mathrm{d}y\
|\ \mathcal{T}(s)=x,N(s)=k,N(t)=n,V(0)=v_{0}\\}$
$\displaystyle=\int_{s}^{t}P\\{\mathcal{T}_{w}(t)\in\mathop{}\\!\mathrm{d}y-x-
v_{k}(w-s)\ |\ N(w)=k+1,N(s)=k,N(t)=n,V(w)=v_{k+1}\\}$ $\displaystyle\hskip
28.45274pt\times P\\{T_{k+1}\in\mathop{}\\!\mathrm{d}w\ |\
N(s)=k,N(t)=n,V(s)=v_{k}\\}$
$\displaystyle=\int_{s}^{t}P\\{\mathcal{T}(t-w)\in\mathop{}\\!\mathrm{d}y-x-
v_{k}(w-s)\ |\ N(t-w)=n-k-1,V(0)=v_{k+1}\\}$ $\displaystyle\hskip
28.45274pt\times P\\{T_{1}\in\mathop{}\\!\mathrm{d}w-s\ |\
N(t-s)=n-k,V(0)=v_{k}\\}$ $\displaystyle=\int_{0}^{t-s}P\\{\mathcal{T}(t-s-
t_{1})\in\mathop{}\\!\mathrm{d}y-x-v_{k}t_{1}\ |\ N(t-s-
t_{1})=n-k-1,V(0)=v_{k+1}\\}$ $\displaystyle\hskip 28.45274pt\times
P\\{T_{1}\in\mathop{}\\!\mathrm{d}t_{1}\ |\ N(t-s)=n-k,V(0)=v_{k}\\}.$ (A.1)

The third equality follows from Markovianity of $N$ and the considerations on
$\mathcal{T}_{w}$. The probability of the telegraph process in the integral of
(A.1) is $0$ if
$t_{1}\not\in\bigl{(}0,\frac{|v_{k+1}(t-s)-(y-x)|}{a_{1}-a_{2}}\bigr{)}$,
which follows since the density in the integral is positive if $\,a_{2}(t-s-
t_{1})<y-x-v_{k}t_{1}<a_{1}(t-s-
t_{1})\implies\begin{cases}t_{1}(v_{k}-a_{2})<-a_{2}(t-s)+(y-x),\\\
t_{1}(a_{1}-v_{k})<a_{1}(t-s)-(y-x).\end{cases}$ By replacing $v_{k}=a_{1}$
and $v_{k}=a_{2}$, simple algebra leads to the condition
$0<t_{1}<\frac{|v_{k+1}(t-s)-(y-x)|}{a_{1}-a_{2}}<t-s$. Hence, we have that

$\displaystyle P\\{$ $\displaystyle\mathcal{T}(t)\in\mathop{}\\!\mathrm{d}y\
|\ \mathcal{T}(s)=x,N(s)=k,N(t)=n,V(0)=v_{0}\\}$ (A.2)
$\displaystyle=\int_{0}^{\frac{|v_{k+1}(t-s)-(y-x)|}{a_{1}-a_{2}}}P\\{T_{1}\in\mathop{}\\!\mathrm{d}t_{1}\
|\ N(t-s)=n-k,V(0)=v_{k}\\}$ $\displaystyle\hskip 34.14322pt\times
P\\{\mathcal{T}(t-s-t_{1})\in\mathop{}\\!\mathrm{d}y-x-v_{k}t_{1}\ |\ N(t-s-
t_{1})=n-k-1,V(0)=v_{k+1}\\}$
$\displaystyle=P\\{\mathcal{T}(t-s)\in\mathop{}\\!\mathrm{d}y-x\ |\
N(t-s)=n-k,\ V(0)=v_{k}\\},$

where the last equality follows by means of a recurrence argument on the
distribution of the telegraph process based on (3.1). Finally, with (A.2) at
hand, (3.2) follows from the law of total probability.

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