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GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 2, 1999, 169-178

ON THE VECTOR PROCESS OBTAINED BY ITERATED
INTEGRATION OF THE TELEGRAPH SIGNAL
ENZO ORSINGHER

Abstract. We analyse the vector process (X0 (t), X1 (t), . . . , Xn (t),
t > 0) where Xk (t) =

Rt

Xk−1 (s)ds, k = 1, . . . , n, and X0 (t) is the

0

two-valued telegraph process.
In particular, the hyperbolic equations governing the joint distributions of the process are derived and analysed.
Special care is given to the case of the process (X0 (t), X1 (t),
X2 (t), t > 0) representing a randomly accelerated motion where some
explicit results on the probability distribution are derived.

1. General Results Concerning the Integrated Telegraph
Signal
Let us consider the two-valued telegraph process
X0 (t) = X(0)(−1)N (t) ,

(1.1)

where X(0) is a random variable which is independent of N (t) and takes
values ±a with equal probability. By N (t) we denote the number of events
of a homogeneous Poisson process up to time t (with rate λ).
Let also
Z t
X0 (s) ds
(1.2)
X1 (t) =
0

and, in general,

Xk (t) =

Z

t

Xk−1 (s) ds,

k = 1, 2, . . . , n.

(1.3)

0

When n = 2, the vector process (X0 (t), X1 (t), X2 (t), t ≥ 0) has a straightforward physical interpretation. Indeed, X2 (t) represents the position of
1991 Mathematics Subject Classification. 60K40.
Key words and phrases. Telegraph process, Poisson process, accelerated motion.

169
c 1999 Plenum Publishing Corporation
1072-947X/99/0300-0169$15.00/0 

170

ENZO ORSINGHER

a particle with acceleration X0 and velocity X1 . The same interpretation is possible for the triple (Xj (t), Xj+1 (t), Xj+2 (t), t ≥ 0), where j =
0, 1, . . . , n − 2.
Most of the current literature (see [1–4]) concerns the process (X0 (t),
X1 (t), t ≥ 0) and our effort here is to examine the general situation, with
special attention to the case n = 2.
Let us now introduce
(
F (x1 , . . . , xn , t) = Pr{X1 (t) ≤ x1 , . . . , Xn (t) ≤ xn , X0 (t) = a},
(1.4)
B(x1 , . . . , xn , t) = Pr{X1 (t) ≤ x1 , . . . , Xn (t) ≤ xn , X0 (t) = −a}
and denote by f = f (x1 , . . . , xn , t), b = b(x1 , . . . , xn , t) the corresponding
densities. Our first result is the differential system governing f and b.
Theorem 1.1. The densities f and b are solutions of

n

X

∂f
 ∂f = −a ∂f −

xj−1
+ λ(b − f ),

 ∂t
∂x1 j=2
∂xj
n
X
 ∂b
∂b
∂b


xj−1
=

a
−
+ λ(f − b).

 ∂t
∂x1
∂xj

(1.5)

j=2

Proof. For the sake of simplicity, we divide the time domain into intervals
of length ∆t and assume that changes of the process X0 occur only at the
endpoints of these intervals. The effect of this assumption disappears as
∆t → 0.
In order to be at (x1 , . . . , xn ) at time t + ∆t one of the following cases
need to occur:
(i) ∆N = 0 and the starting point for each process must be
xk = xk − xk−1 ∆t + · · · + (−1)j xk−j

(∆t)j
(∆t)k
+ · · · + (−1)k a
,
j!
k!

where k = 1, 2, . . . , n and x0 = 0.
(ii) ∆N = 1 and the starting point for each process is
xk = xk − xk−1 ∆t + · · · + (−1)j xk−j

(∆t)k
(∆t)j
+ · · · + (−1)k−1 a
.
j!
k!

(iii) ∆N > 1.

Restricting ourselves only to the first-order terms we have
f (x1 , . . . , xn , t + ∆t) =
= (1 − λ∆t)f (x1 − a∆t, x2 − x1 ∆t, . . . , xn − xn−1 ∆t, t) +

+λ∆tb(x1 + a∆t, x2 − x1 ∆t, . . . , xn − xn−1 ∆t, t) + o(∆t).

(1.6)

VECTOR PROCESS OBTAINED BY ITERATED INTEGRATION

171

Expanding in Taylor series and passing to the limit as ∆t → 0, we obtain
the first equation of system (1.5). An analogous treatment leads to the
second equation of (1.5).
If p = f + b and w = f − b, the above system can be rewritten as

n
X
 ∂p

∂w
∂p


−a
xj−1
=
−
,

 ∂t
∂x1 j=2
∂xj
(1.7)
n

∂w
 ∂w = −a ∂p − X x

− 2λw.


j−1

∂t
∂x1
∂xj
j=2

A rather surprising fact is that the equation governing p (to be obtained
from (1.7)) is of third order for any n ≥ 2.
Theorem 1.2. The probability density p = p(x1 , . . . , xn , t) is a solution
of
n
n
n
 ∂p X
∂p ‘
∂p ‘
∂  ∂p X
∂ h ∂2p X

x
x
x
+2λ
+
+
+
+
j−1
r−1
j−1
∂x1 ∂t2 j=2
∂xj ∂t r=2
∂xr
∂t j=2
∂xj

+

n

X
j=2

xj−1

n
∂2p
∂p ‘
∂2p i
∂  ∂p X
.
+
xj−1
− a2 2 = −
∂t∂xj
∂x1
∂x2 ∂t j=2
∂xj

(1.8)

Proof. Deriving the first equation of (1.7) with respect to time t and inserting the other one derived with respect to x1 , we obtain
n

X
∂2p
∂2w
∂2p
x
=
−a
=
−
j−1
∂t2
∂x1 ∂t j=2
∂t∂xj
• X
”
n
n
X
∂2p
∂2w
∂2p
∂w
∂w
xj−1
− 2λ
xj−1
−
= −a − a 2 −
−
=
∂x1
∂x2 j=2
∂xj ∂x1
∂x1
∂t∂xj
j=2
’
“
n
n
2
X
∂p
∂
∂w
∂p X
2 ∂ p
xr−1
=a
+a
xj−1
−
+
−
∂x21
∂x2 j=2
∂xj ∂t r=2
∂xr
“ X
’
n
n
∂p
∂p X
∂2p
−
+
.
xj−1
−2λ
xj−1
∂t j=2
∂xj
∂t∂xj
j=2
A further derivation with respect to x1 then leads to equation (1.8).
Dealing with the third-order equation (1.8) implies substantial difficulties. In order to circumvent them we present the second-order equations
governing the densities f and b.

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ENZO ORSINGHER

For convenience we write f = e−λt f , b = e−λt b and obtain from (1.5)

n
X

∂f
∂f
∂f


−
+ λb,
=
−a
xj−1

 ∂t
∂x1 j=2
∂xj
(1.9)
n
X

∂b
∂b
∂b


=a
−
+ λf .
xj−1


∂t
∂x1
∂xj
j=2

Some calculations now suffice to obtain from the above system the secondorder equations
 2
n
2
X
∂f
∂2f
∂ f

2 ∂ f


a
=
−2
+a
−
+
x
j−1

2
2

∂xj ∂t
∂x1
∂x2
 ∂t
j=2


’X
“

n
n

X

∂f
∂


+ λ2 f ,
xr−1
xj−1
−


∂x
∂x
j
r
r=2
j=2
(1.10)
n
2
X

∂2b
∂2b
∂b

2 ∂ b

a
+
a
−
x
=
−2
−
 2
j−1

∂t
∂xj ∂t
∂x21
∂x2


j=2


’
“
n
n

X
X

∂
∂b


xj−1
xr−1
+ λ2 b.
−


∂x
∂x
j
r
r=2
j=2

Remark 1.1. If λ → ∞ and a → ∞ in such a way that
from (1.8) the equation

a2
λ

→ 1, we obtain

n

∂p X
∂p
1 ∂2p
xj−1
+
=
,
∂t j=2
∂xj
2 ∂x21

(1.11)

which is satisfied by the probability law of the vector process (X1 (t), . . . ,
Xn (t), t ≥ 0), where X1 is a standard Brownian motion and
Z t
Xk−1 (s) ds, k = 2, . . . , n.
Xk (t) =
0

2. The Special Case n = 2
A deeper analysis is possible in the case of the vector process (X0 (t),
X1 (t), X2 (t), t ≥ 0) representing a uniformly accelerated random motion.
In that case system (1.6) reads as

∂f
∂f
∂f


− x1
+ λ(b − f ),
= −a
∂t
∂x1
∂x2
(2.1)
∂b
∂b
∂b


=a
− x1
+ λ(f − b)
∂t
∂x1
∂x2

VECTOR PROCESS OBTAINED BY ITERATED INTEGRATION

and equations (1.10) are reduced to the form
 2
2
2
2

 ∂ f = −2x1 ∂ f − x21 ∂ f + a2 ∂ f + a ∂f + λ2 f ,

2
2
∂t
∂t∂x2
∂x2
∂x21
∂x2
2
2
2
2
∂
b
b
b
b
∂
∂
∂
∂b

2
2


−a
= −2x1
− x1 2 + a
+ λ2 b.
∂t2
∂t∂x2
∂x2
∂x21
∂x2

173

(2.2)

By combining equations (2.2) it is easy to obtain (1.8) once again provided
that functions f and b are inserted.
In our view the most interesting result concerning equations (2.2) is given
in the next theorem.

Theorem 2.1. The function

1
a2 t2 − x12
a2 t2 − x12 ‘
1
(2.3)
, x2 − x 1 t −
f (x1 , x2 , t) = q x2 − x1 t +
2
4a
2
4a
is a solution of the first equation of (2.2), provided that q = q(u, w) is a
solution of
(w − u)

∂q
λ2
∂2q
q.
=
+
∂u∂w
∂w 2a

(2.4)

Analogously,

1
a2 t2 − x21
a2 t2 − x12 ‘
1
, x2 − x 1 t −
(2.5)
b(x1 , x2 , t) = g x2 − x1 t +
2
4a
2
4a
is a solution of the second equation of (2.2) provided that g is a solution of
∂2g
λ2
∂g
=−
+
g.
(2.6)
∂u∂w
∂w 2a
Proof. Since only simple calculations are involved, we omit the details.
(w − u)

Remark 2.1. It is interesting
√ that equations (2.4) and (2.6) are reduced
by the transformation z = w − u to the Bessel equation
∂2q
1 ∂q
2λ2
+
+
q = 0.
∂z 2
z ∂z
a

(2.7)

This result is due to the fact that q = q(s) is a function depending only
on x1 through z. This is related to the well-known fact (see [1]) that the
marginals
Z
Z
f (x1 , x2 , t) dx2

and

b(x1 , x2 , t) dx2

are expressed in terms of Bessel p
functions of order zero with imaginary
arguments and depending on z = a2 t2 − x21 .
To increase our insight into the vector process (X1 (t), X2 (t), t > 0) we
first note that possible values of X1 at time t are within [−at, at] and possible

174

ENZO ORSINGHER

values of X2 are located in the interval [− 21 at2 , 12 at2 ]. However, not all
couples of the set
n
o
1
1
R = x1 , x2 : −at ≤ x1 ≤ at, − at2 ≤ x2 ≤ at2
2
2
can be occupied. For example, it is impossible for the process X1 to take
values close to −at and for X2 to occupy positions near 21 at2 (the interpretation of X1 as velocity and X2 as the current position of a moving particle
can help here).
We now present the following result.
Theorem 2.2. At time t the support of (X1 (t), X2 (t)) is the set
n
a2 t2 − x21
1
≤ x2 ≤
S = x1 , x2 : −at ≤ x1 ≤ at, x1 t −
2
4a
o
1
a2 t2 − x12
≤ x1 t +
,
(2.8)
2
4a
which can be rewritten as
n
p
1
1
S = x1 , x2 : − at2 ≤ x2 ≤ at2 , at − 2a2 t2 − 4ax2 ≤ x1 ≤
2
2
o
p
(2.9)
≤ −at + 2a2 t2 + 4ax2 .
Proof. Assume that at time t, X1 (t) = x1 . If
Z t
T+ =
I{X0 (s)>0} ds = meas{s < t : X0 (s) > 0},
0
Z t
I{X0 (s) t2 > t1 + at−x
2a .
t3

distribution (2.16) follows from the above results.

On the basis of all the previous results we can present the following
explicit formulas.
Theorem 2.5.
Œ
‰
ˆ
dx2
Pr X2 (t) ∈ dx2 Œ X1 (t) = x1 , N (t) = 2 =
, Bi < x2 < Bs ,
Bs − Bi
Œ
ˆ
‰
Pr X2 (t) ∈ dx2 Œ X1 (t) = x1 , N (t) = 3 =

x2 − Bi ) ‘
dx2
log 1 −
, Bi < x2 < Bs .
=−
B s − Bi
Bs − Bi

Proof. From the first formula of (2.12) we readily have
Œ
‰
ˆ
Pr X2 (t) < x2 ΠX1 (t) = x1 , N (t) = 2 =

n
o
x2 − Bi ŒŒ
2a
= Pr T1 <
X1 (t) = x1 , N (t) = 2 =
at − x1
at + x1

where the last step is justified by (2.15).
The derivation of (2.18) requires some additional details.
Taking into account the second formula of (2.12), we have
Œ
ˆ
‰
Pr X2 (t) < x2 ΠX1 (t) = x1 , N (t) = 3 =

(2.17)

(2.18)

x2 −Bi
at−x1

Z
0

dt1 ,

178

ENZO ORSINGHER

n
= Pr T2 < T1 +
=

o
Œ
Bs − x2
ΠX1 (t) = x1 , N (t) = 3 =
x1 + at − 2T2
ZZ
4a2
dt1 dt2 =
2
a t2 − x21
x −B

2
i
0