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 (X_{0}(t), X_{1}(t), …, X_{n}(t),

t > 0) where X_{k}(t) = ∫_{0}^{t} X_{k−1}(s)ds, k = 1, …, n, and X_{0}(t) is the

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 (X_{0}(t), X_{1}(t), X_{2}(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

X_{0}(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

X_{1}(t) = ∫_{0}^{t} X_{0}(s) ds  (1.2)

and, in general,

X_{k}(t) = ∫_{0}^{t} X_{k−1}(s) ds, k = 1, 2, …, n.  (1.3)

When n = 2, the vector process (X_{0}(t), X_{1}(t), X_{2}(t), t ≥ 0) has a straightforward physical interpretation. Indeed, X_{2}(t) represents the position of

169 a particle with acceleration X_{0} and velocity X_{1}. The same interpretation is possible for the triple (X_{j}(t), X_{j+1}(t), X_{j+2}(t), t ≥ 0), where j = 0, 1, …, n − 2.

1991 Mathematics Subject Classification. 60K40.

Key words and phrases. Telegraph process, Poisson process, accelerated motion.

1072-947X/99/0300-0169$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

Most of the current literature (see [1–4]) concerns the process (X_{0}(t), X_{1}(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

\begin{cases}F (x_{1}, …, x_{n}, t) = Pr\{X_{1}(t) ≤ x_{1}, …, X_{n}(t) ≤ x_{n}, X_{0}(t) = a\}, \\ B(x_{1}, …, x_{n}, t) = Pr\{X_{1}(t) ≤ x_{1}, …, X_{n}(t) ≤ x_{n}, X_{0}(t) = −a\} \end{cases} (1.4)

and denote by f = f(x_{1}, …, x_{n}, t), b = b(x_{1}, …, x_{n}, 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

\begin{cases}\frac{∂f}{∂t} = −a \frac{∂f}{∂x_{1}} − ∑_{j=2}^{n} x_{j−1} \frac{∂f}{∂x_{j}} + λ(b − f), \\ \frac{∂b}{∂t} = a \frac{∂b}{∂x_{1}} − ∑_{j=2}^{n} x_{j−1} \frac{∂b}{∂x_{j}} + λ(f − b). \end{cases}  (1.5)

Proof. For the sake of simplicity, we divide the time domain into intervals of length Δt and assume that changes of the process X_{0} occur only at the endpoints of these intervals. The effect of this assumption disappears as Δt → 0.

In order to be at (x_{1}, …, x_{n}) at time t + Δt one of the following cases need to occur:

(i) ΔN = 0 and the starting point for each process must be

\overline{x}_{k} = x_{k} − x_{k−1}Δt + ⋯ + (−1)^{j}x_{k−j} \frac{(Δt)^{j}}{j!} + ⋯ + (−1)^{k}a \frac{(Δt)^{k}}{k!} ,

where k = 1, 2, …, n and x_{0} = 0.

(ii) ΔN = 1 and the starting point for each process is

\overline{x}_{k} = x_{k} − x_{k−1}Δt + ⋯ + (−1)^{j}x_{k−j} \frac{(Δt)^{j}}{j!} + ⋯ + (−1)^{k−1}a \frac{(Δt)^{k}}{k!} .

(iii) ΔN > 1.

Restricting ourselves only to the first-order terms we have

f(x_{1}, …, x_{n}, t + Δt) =
= (1 − λΔt)f(x_{1} − aΔt, x_{2} − x_{1}Δt, …, x_{n} − x_{n−1}Δt, t) +
+λΔtb(x_{1} + aΔt, x_{2} − x_{1}Δt, …, x_{n} − x_{n−1}Δt, t) + o(Δt). (1.6)

Expanding in Taylor series and passing to the limit as Δt → 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

\begin{cases}\frac{∂p}{∂t} = −a \frac{∂w}{∂x_{1}} − ∑_{j=2}^{n} x_{j−1} \frac{∂p}{∂x_{j}} , \\ \frac{∂w}{∂t} = −a \frac{∂p}{∂x_{1}} − ∑_{j=2}^{n} x_{j−1} \frac{∂w}{∂x_{j}} − 2λw. \end{cases}  (1.7)

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(x_{1}, …, x_{n}, t) is a solution of

\frac{∂}{∂x_{1}} [\frac{∂^{2}p}{∂t^{2}} +∑_{j=2}^{n} x_{j−1} \frac{∂}{∂x_{j}} (\frac{∂p}{∂t} +∑_{r=2}^{n} x_{r−1} \frac{∂p}{∂x_{r}} )+2λ(\frac{∂p}{∂t} +∑_{j=2}^{n} x_{j−1} \frac{∂p}{∂x_{j}} )+

+ ∑_{j=2}^{n} x_{j−1} \frac{∂^{2}p}{∂t∂x_{j}} − a^{2} \frac{∂^{2}p}{∂x_{1}^{2}} ] = −\frac{∂}{∂x_{2}} (\frac{∂p}{∂t} + ∑_{j=2}^{n} x_{j−1} \frac{∂p}{∂x_{j}} ). (1.8)

Proof. Deriving the first equation of (1.7) with respect to time t and inserting the other one derived with respect to x_{1}, we obtain

\frac{∂^{2}p}{∂t^{2}} = −a \frac{∂^{2}w}{∂x_{1}∂t} − ∑_{j=2}^{n} x_{j−1} \frac{∂^{2}p}{∂t∂x_{j}} =

= −a[ − a \frac{∂^{2}p}{∂x_{1}^{2}} − \frac{∂w}{∂x_{2}} − ∑_{j=2}^{n} x_{j−1} \frac{∂^{2}w}{∂x_{j}∂x_{1}} − 2λ \frac{∂w}{∂x_{1}} ] − ∑_{j=2}^{n} x_{j−1} \frac{∂^{2}p}{∂t∂x_{j}} =

= a^{2} \frac{∂^{2}p}{∂x_{1}^{2}} + a \frac{∂w}{∂x_{2}} − ∑_{j=2}^{n} x_{j−1} \frac{∂}{∂x_{j}} (\frac{∂p}{∂t} + ∑_{r=2}^{n} x_{r−1} \frac{∂p}{∂x_{r}} ) −

−2λ(\frac{∂p}{∂t} + ∑_{j=2}^{n} x_{j−1} \frac{∂p}{∂x_{j}} ) − ∑_{j=2}^{n} x_{j−1} \frac{∂^{2}p}{∂t∂x_{j}} .

A further derivation with respect to x_{1} 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.

For convenience we write f = e^{−λt}\overline{f}, b = e^{−λt}\overline{b} and obtain from (1.5)

\begin{cases}\frac{∂\overline{f}}{∂t} = −a \frac{∂\overline{f}}{∂x_{1}} − ∑_{j=2}^{n} x_{j−1} \frac{∂\overline{f}}{∂x_{j}} + λ\overline{b}, \\ \frac{∂\overline{b}}{∂t} = a \frac{∂\overline{b}}{∂x_{1}} − ∑_{j=2}^{n} x_{j−1} \frac{∂\overline{b}}{∂x_{j}} + λ\overline{f}. \end{cases}  (1.9)

Some calculations now suffice to obtain from the above system the secondorder equations

\begin{cases}\frac{∂^{2}\overline{f}}{∂t^{2}} = −2 ∑_{j=2}^{n} x_{j−1} \frac{∂^{2}\overline{f}}{∂x_{j}∂t} + a^{2} \frac{∂^{2}\overline{f}}{∂x_{1}^{2}} + a \frac{∂\overline{f}}{∂x_{2}} − \\ − ∑_{j=2}^{n} x_{j−1} \frac{∂}{∂x_{j}} ( ∑_{r=2}^{n} x_{r−1} \frac{∂\overline{f}}{∂x_{r}} ) + λ^{2}\overline{f}, \\ \frac{∂^{2}\overline{b}}{∂t^{2}} = −2 ∑_{j=2}^{n} x_{j−1} \frac{∂^{2}\overline{b}}{∂x_{j}∂t} + a^{2} \frac{∂^{2}\overline{b}}{∂x_{1}^{2}} − a \frac{∂\overline{b}}{∂x_{2}} − \\ − ∑_{j=2}^{n} x_{j−1} \frac{∂}{∂x_{j}} ( ∑_{r=2}^{n} x_{r−1} \frac{∂\overline{b}}{∂x_{r}} ) + λ^{2}\overline{b}. \end{cases}  (1.10)

Remark 1.1. If λ → ∞ and a → ∞ in such a way that \frac{a^{2}}{λ} → 1, we obtain from (1.8) the equation

\frac{∂p}{∂t} + ∑_{j=2}^{n} x_{j−1} \frac{∂p}{∂x_{j}} = \frac{1}{2} \frac{∂^{2}p}{∂x_{1}^{2}} ,  (1.11)

which is satisfied by the probability law of the vector process (X_{1}(t), …, X_{n}(t), t ≥ 0), where X_{1} is a standard Brownian motion and

X_{k}(t) = ∫_{0}^{t} X_{k−1}(s) ds, k = 2, …, n.

2. The Special Case n = 2

A deeper analysis is possible in the case of the vector process (X_{0}(t), X_{1}(t), X_{2}(t), t ≥ 0) representing a uniformly accelerated random motion.

In that case system (1.6) reads as

\begin{cases}\frac{∂f}{∂t} = −a \frac{∂f}{∂x_{1}} − x_{1} \frac{∂f}{∂x_{2}} + λ(b − f), \\ \frac{∂b}{∂t} = a \frac{∂b}{∂x_{1}} − x_{1} \frac{∂b}{∂x_{2}} + λ(f − b) \end{cases}  (2.1)

and equations (1.10) are reduced to the form

\begin{cases}\frac{∂^{2}\overline{f}}{∂t^{2}} = −2x_{1} \frac{∂^{2}\overline{f}}{∂t∂x_{2}} − x_{1}^{2} \frac{∂^{2}\overline{f}}{∂x_{2}^{2}} + a^{2} \frac{∂^{2}\overline{f}}{∂x_{1}^{2}} + a \frac{∂\overline{f}}{∂x_{2}} + λ^{2}\overline{f}, \\ \frac{∂^{2}\overline{b}}{∂t^{2}} = −2x_{1} \frac{∂^{2}\overline{b}}{∂t∂x_{2}} − x_{1}^{2} \frac{∂^{2}\overline{b}}{∂x_{2}^{2}} + a^{2} \frac{∂^{2}\overline{b}}{∂x_{1}^{2}} − a \frac{∂\overline{b}}{∂x_{2}} + λ^{2}\overline{b}. \end{cases}  (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

\overline{f}(x_{1}, x_{2}, t) = q(x_{2} − \frac{1}{2} x_{1}t + \frac{a^{2}t^{2} − x_{1}^{2}}{4a} , x_{2} − \frac{1}{2} x_{1}t − \frac{a^{2}t^{2} − x_{1}^{2}}{4a} ) (2.3)

is a solution of the first equation of (2.2), provided that q = q(u, w) is a solution of

(w − u) \frac{∂^{2}q}{∂u∂w} = \frac{∂q}{∂w} + \frac{λ^{2}}{2a} q.  (2.4)

Analogously,

\overline{b}(x_{1}, x_{2}, t) = g(x_{2} − \frac{1}{2} x_{1}t + \frac{a^{2}t^{2} − x_{1}^{2}}{4a} , x_{2} − \frac{1}{2} x_{1}t − \frac{a^{2}t^{2} − x_{1}^{2}}{4a} ) (2.5)

is a solution of the second equation of (2.2) provided that g is a solution of

(w − u) \frac{∂^{2}g}{∂u∂w} = −\frac{∂g}{∂w} + \frac{λ^{2}}{2a} g.  (2.6)

Proof. Since only simple calculations are involved, we omit the details.

Remark 2.1. It is interesting that equations (2.4) and (2.6) are reduced by the transformation z = \sqrt{w − u} to the Bessel equation

\frac{∂^{2}q}{∂z^{2}} + \frac{1}{z} \frac{∂q}{∂z} + \frac{2λ^{2}}{a} q = 0.  (2.7)

This result is due to the fact that q = q(s) is a function depending only on x_{1} through z. This is related to the well-known fact (see [1]) that the marginals

∫ f(x_{1}, x_{2}, t) dx_{2} and ∫ b(x_{1}, x_{2}, t) dx_{2}

are expressed in terms of Bessel functions of order zero with imaginary arguments and depending on z = \sqrt{a^{2}t^{2} − x_{1}^{2}}.

To increase our insight into the vector process (X_{1}(t), X_{2}(t), t > 0) we first note that possible values of X_{1} at time t are within [−at, at] and possible values of X_{2} are located in the interval [−\frac{1}{2}at^{2}, \frac{1}{2}at^{2}]. However, not all couples of the set

R = \{x_{1}, x_{2} : −at ≤ x_{1} ≤ at, −\frac{1}{2} at^{2} ≤ x_{2} ≤ \frac{1}{2} at^{2}\}

can be occupied. For example, it is impossible for the process X_{1} to take values close to −at and for X_{2} to occupy positions near \frac{1}{2}at^{2} (the interpretation of X_{1} as velocity and X_{2} 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 (X_{1}(t), X_{2}(t)) is the set

S = \{x_{1}, x_{2} : −at ≤ x_{1} ≤ at, \frac{1}{2} x_{1}t − \frac{a^{2}t^{2} − x_{1}^{2}}{4a} ≤ x_{2} ≤

≤ \frac{1}{2} x_{1}t + \frac{a^{2}t^{2} − x_{1}^{2}}{4a} \},  (2.8)

which can be rewritten as

S = \{x_{1}, x_{2} : −\frac{1}{2}at^{2} ≤ x_{2} ≤ \frac{1}{2}at^{2}, at − \sqrt{2a^{2}t^{2} − 4ax_{2}} ≤ x_{1} ≤

≤ −at + \sqrt{2a^{2}t^{2} + 4ax_{2}}\}.  (2.9)

Proof. Assume that at time t, X_{1}(t) = x_{1}. If

T_{+} = ∫_{0}^{t} I_{{X_{0}(s)>0}} ds = meas{s < t : X_{0}(s) > 0},
T_{−} = ∫_{0}^{t} I_{{X_{0}(s)<0}} ds = meas{s < t : X_{0}(s) < 0},

it is clear that a(T_{+}−T_{−}) = x_{1}. In that case the farthest position to the right of the origin is reached when the entire rightward motion occurs initially, during time T_{+}.

The final position is

max X_{2} = \frac{1}{2} aT_{+}^{2} + aT_{+}T_{−} − \frac{1}{2} aT_{−}^{2}.

Since T_{+} + T_{−} = t, we obtain

max X_{2} = \frac{1}{2} x_{1}t + \frac{a^{2}t^{2} − x_{1}^{2}}{4a} .

Conversely, min X_{2} is reached when the leftward motion is performed initially.

In writing down (2.9), the sign must be chosen in such a way that for x_{2} = ±\frac{1}{2}at^{2} we should have x_{1} = ±at.

All the information which can be read from the form of S coincides with the intuition. In particular, when x_{1} = ±at we have x_{2} = \frac{1}{2}at^{2} and the closer x_{1} is to the origin, the bigger the interval of possible values of X_{2} becomes.

We note that, if X_{1}(t) = x_{1}, at time t the random variable X_{2}(t) can take any value from the interval [\frac{1}{2}x_{1}t − \frac{a^{2}t^{2}−x_{1}^{2}}{4a} , \frac{1}{2}x_{1}t + \frac{a^{2}t^{2}−x_{1}^{2}}{4a}].

To realize this we present

Theorem 2.3. If N(t) = m, X_{0}(0) = a, T_{i} is the random time at which the i-th event of the driving Poisson process (i ≤ m) occurs, then

\begin{cases}X_{1}(t) = 2a ∑_{i=1}^{m}(−1)^{i−1}T_{i} + a(−1)^{m}t, \\ X_{2}(t) = a ∑_{i=1}^{m}(−1)^{i}T_{i}^{2} + 2at ∑_{i=1}^{m}(−1)^{i−1}T_{i} + (−1)^{m} \frac{1}{2} at^{2}. \end{cases} (2.10)

Proof. At the instant the m-th Poisson event takes place, the processes (X_{1}(t), X_{2}(t), t ≥ 0) take values X_{1}(T_{m}) and X_{2}(T_{m}) and thus, after some substitutions and simplifications, at t < T_{m} we have (by induction)

X_{2}(t) = X_{2}(T_{m}) + X_{1}(T_{m})(t − T_{m}) + \frac{1}{2} a(−1)^{m}(t − T_{m})^{2} =

= a ∑_{i=1}^{m}(−1)^{i}T_{i}^{2} + 2at ∑_{i=1}^{m}(−1)^{i−1}T_{i} + (−1)^{m} \frac{1}{2} at^{2}.

Corollary 2.1. It N(t) = m, X_{1}(t) = x_{1}, X_{0}(0) = a, possible positions which can be occupied at time t are given by

X_{2}(t) = a ∑_{i=1}^{m−1}(−1)^{i}T_{i}^{2} + \frac{(−1)^{m}}{4a} (x_{1} − a(−1)^{m}t +

+ 2a ∑_{i=1}^{m−1}(−1)^{i−1}T_{i})^{2} + tx_{1} − \frac{1}{2} (−1)^{m}at^{2}.  (2.11)

Proof. From the first equation of (2.10) we get

T_{m} = \frac{x_{1} − a(−1)^{m}t − 2a ∑_{i=1}^{m−1}(−1)^{i−1}T_{i}}{2a(−1)^{m}}

and rewriting the second equation as,

X_{2}(t) = a ∑_{i=1}^{m−1}(−1)^{i}T_{i}^{2} + a(−1)^{m}T_{m}^{2} + t(x_{1} − a(−1)^{m}t) + \frac{1}{2} (−1)^{m}at^{2},

after a substitution the desired result emerges.

Remark 2.2. If N(t) = m, X_{0}(0) = a, and the random times T_{1}, …, T_{m−1} are fixed, possible couples (x_{1}, x_{2}) of velocities and positions form a parabola.

Remark 2.3. If X_{2}(t) = x_{2}, B_{i} = \frac{1}{2}x_{1}t − \frac{a^{2}t^{2}−x_{1}^{2}}{4a}, B_{s} = \frac{1}{2}x_{1}t + \frac{a^{2}t^{2}−x_{1}^{2}}{4a}, then from (2.11) we derive the special cases.

\begin{aligned}m = 2 & x_{2} = T_{1}(at − x_{1}) + B_{i}, \\ m = 3 & x_{2} = B_{s} + (T_{1} − T_{2})(x_{1} + at − 2aT_{1}), \\ m = 4 & x_{2} = B_{i} + (T_{2} − T_{1})(2aT_{2} − (at − x_{1})) + \\ & + T_{3}(at − x_{1} + 2a(T_{1} − T_{2})). \end{aligned}  (2.12)

These formulas permit us to show that, if N(t) ≥ 2, at time t for fixed values of X_{1}(t), possible positions (namely the values of X_{2}(t)) cover the whole interval [B_{i}, B_{s}].

It must be observed that various curves formed by the couples (x_{1}, x_{2}) must be analysed taking into account the constraints concerning times T_{i}.

For example, if m = 2, we have 0 ≤ T_{1} ≤ \frac{at+x_{1}}{2a}. Thus if T_{1} = 0, then x_{2} = B_{i} and the couples (x_{1}, x_{2}) form one of the parabolas bounding the set S.

Analogously, if T_{1} = \frac{at+x_{1}}{2a}, then x_{2} = B_{s} and the other curve bounding S is obtained.

If T_{1} = \frac{at+x_{1}}{4a}, we obtain the line of symmetry of S, whose equation is x_{2} = \frac{1}{2}x_{1}t.

Remark 2.4. The direct calculation of

Pr {X_{2}(t) ∈ dx_{2} | X_{0}(0) = a, X_{1}(t) = x_{1}, N(t) = m}  (2.13)

presents considerable difficulties even when m is small.

We have been able to derive these results when m = 2, 3. From the formulas obtained it follows that general expressions of the distribution densities of (X_{1}(t), X_{2}(t), t ≥ 0) are of form (2.3) and (2.5) and are solutions of equations (2.4) and (2.6).

To evaluate (2.13) it is necessary to know the conditional distribution:

Pr {T_{1} ∈dt_{1}, …, T_{m−1} ∈dt_{m−1}|N(t)=m, X_{1}(t)=x_{1}, X_{0}(0)=a}. (2.14)

In particular, we have obtained

Theorem 2.4.

Pr {T_{1} ∈ dt_{1} | X_{1}(t) = x_{1}, N(t) = 2, X_{0}(0) = a} = 2a/(at + x_{1})

when 0 < t_{1} < (at + x_{1})/2a,  (2.15)

Pr \{T_{1} ∈ dt_{1}, T_{2} ∈ dt_{2} | X_{1}(t) = x_{1}, N(t) = 3, X_{0}(0) = a\} = \frac{4a^{2}}{a^{2}t^{2} − x_{1}^{2}}

when 0 < t_{1} < (at + x_{1})/2a and t_{1} < t_{2} < t_{1} + (at − x_{1})/2a. (2.16)

Proof. When m = 3, from (2.12) we immediately have

Pr {N(t) = 3, X_{1}(t) ≤ x_{1}, X_{0}(0) = a} =

= \frac{3!}{t^{3}} \{ ∫_{0}^{(at+x_{1})/2a}dt_{1} ∫_{t_{1}}^{t_{1}+(at−x_{1})/2a}dt_{2} ∫_{t_{2}}^{t_{2}−t_{1}+(at+x_{1})/2a}dt_{3} +

+ ∫_{0}^{(at+x_{1})/2a}dt_{1} ∫_{t_{1}+(at−x_{1})/2a}^{t} dt_{2} ∫_{t_{2}}^{t} dt_{3}\} = \frac{(at + x_{1})^{2}(2at − x_{1})}{(2a)^{2}t^{3}a} .

Furthermore, when 0 < t_{1} < \frac{at+x_{1}}{2a}, we have

Pr {T_{1} ∈ dt_{1}, T_{2} ∈ dt_{2} | X_{1}(t) = x_{1}, N(t) = 3, X_{0}(0) = a} =

= \begin{cases}\frac{3!}{t^{3}} (\frac{at + x_{1}}{2a} − t_{1})dt_{1}dt_{2} & if t_{1} < t_{2} < t_{1} + \frac{at−x_{1}}{2a} , \\ \frac{3!}{t^{3}} (t − t_{2}) dt_{1} dt_{2} & if t > t_{2} > t_{1} + \frac{at−x_{1}}{2a} . \end{cases}

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.

Pr \{X_{2}(t)∈dx_{2} | X_{1}(t)=x_{1}, N(t)=2\}= \frac{dx_{2}}{B_{s} − B_{i}} , B_{i} <x_{2} <B_{s}, (2.17)
Pr {X_{2}(t) ∈ dx_{2} | X_{1}(t) = x_{1}, N(t) = 3} =
= −\frac{dx_{2}}{B_{s} − B_{i}} log (1 − \frac{x_{2} − B_{i})}{B_{s} − B_{i}}), B_{i} < x_{2} < B_{s}.  (2.18)

Proof. From the first formula of (2.12) we readily have

Pr {X_{2}(t) < x_{2} | X_{1}(t) = x_{1}, N(t) = 2} =

= Pr \{T_{1} < \frac{x_{2} − B_{i}}{at − x_{1}} | X_{1}(t) = x_{1}, N(t) = 2\} = \frac{2a}{at + x_{1}} ∫_{0}^{\frac{x_{2}−B_{i}}{at−x_{1}}}dt_{1},

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

Pr {X_{2}(t) < x_{2} | X_{1}(t) = x_{1}, N(t) = 3} =

= Pr \{T_{2} < T_{1} + \frac{B_{s} − x_{2}}{x_{1} + at − 2T_{2}} | X_{1}(t) = x_{1}, N(t) = 3\} =

= ∫∫_{\begin{matrix}0<t_{1}< \frac{x_{2}−B_{i}}{at−x_{1}} \\ t_{1}+ \frac{Bs−x_{2}}{x_{1}+at−2at_{1}} <t_{2}<t_{1}+ \frac{at−x_{1}}{2a} \end{matrix}} \frac{4a^{2}}{a^{2}t^{2} − x_{1}^{2}} dt_{1} dt_{2} =

= \frac{x_{2} − B_{i}}{B_{s} − B_{i}} + \frac{B_{s} − x_{2}}{B_{s} − B_{i}} log (1 − \frac{x_{2} − B_{i}}{B_{s} − B_{i}} ).

Acknowledgement

The author appreciates the possibility to discuss the above results at the conference “Evolutionary Stochastic Systems in Physics and Biology” (1992, Katsiveli, Ukraine) and during his staying at A. Razmadze Mathematical Institute (1996, Tbilisi, Georgia).

References

1. E. Orsingher, Probability law flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff’s laws. Stochastic Process. Appl. 34(1990), 49–66.

2. M. O. Hongler and L. Streit, Generalized master equations and the telegrapher’s equation. Physica A 165(1990), 196–206.

3. A. D. Kolesnik, One-dimensional models of random Markovian evolutions. (Russian) Preprint, 1990.

4. E. Orsingher and B. Bassan, On a 2n-valued telegraph signal and the related integrated process. Stochastics Stochastics Rep. 38(1992), 159– 173.

(Received 25.12.1997)

Author’s address: Dipartimento di Statistica Probabilità e Statistiche Applicate Università degli Studi di Roma “La Sapienza” Piazzale A. Moro 5, 00185 Roma Italy