1. Introduction
Starting from the motion of a particle according to a telegraph process on an ellipse, random harmonic oscillators can be built on the ellipse. These models have many applications in physics and engineering, and they are also studied using Langevin stochastic differential equations with a harmonic potential, as can be seen in
Gitter (
2005). In this survey article, the particular case of colored dichotomous noise is also considered as a driving source.
The Cox process
,
, also known as a doubly stochastic Poisson process, is a generalization of a Poisson process, whose rate
is a positive random variable. The process is named after the statistician David Cox, who first published the model in 1955 (
Cox (
1955);
Pinsky and Karlin (
2011)). We consider in this paper the first new telegraph model, so-called Cox-based telegraph process, and its applications in finance, namely, in option pricing. Our second new model is the so-called elliptical telegraph process. The telegraph process on a circle was considered in
De Gregorio and Iafrate (
2021). Some finite-velocity random motions driven by the geometric counting process were studied in
Di Crescenzo et al. (
2023). Symmetrical and asymmetrical classical telegraph processes and their applications in option pricing and spread option valuations were considered in
Pogorui et al. (
2021a,
2021b,
2021c,
2022).
In this work, we consider a simple Cox process, namely, a Poisson process of rate
, where
is a random variable. We assume that there exists a cumulative distribution function (CDF)
of
; hence, the probability distribution of
is as follows:
We should remark the following properties of
and
Taking into account the well-known conditional variance formula
we have
It is also straightforward to obtain a formula for the conditional probability distribution of
assuming
,
Furthermore, if there exists a PDF
then the conditional probability density function of
given
can be written as
Suppose that a particle moves in a line as follows: It starts from a point at initial time, and moves with absolute speed along directions 1 or −1 with probability until the first Cox switching instant , where , are independent and identically distributed (iid) random variables that have the same CDF and its corresponding PDF , . At instant , the particle can change the direction of movement to the opposite with probability or continues its motion in the same direction with probability , and moves with the absolute random velocity until the next Cox switching instant . At , the particle can change the direction of movement to the opposite with probability or continues its motion in the same direction with probability with the absolute random velocity until the next Cox switching, and so on.
We assume that all and , are mutually independent.
Let us denote by
the position of the particle at time
t. Then, it is easily seen that
with
, and for all
, the random variables
,
are independent.
We will also investigate the telegraph process
on an ellipse centered at the origin (we call it elliptical telegraph process)
i.e., a random motion of an object or particle with constant absolute velocity and a switching Cox process governing the direction of movements on the ellipse.
This elliptical telegraph process
can be represented as
where
is defined in Equation (
2). See
Section 4.
2. Characteristic Function
We assume that the PDF
does exist. Now, let us denote as
the Laplace transform of
From Equation (
1), we conclude that
Let us define the characteristic function of the process by . Thus, we have
Theorem 1. The characteristic function satisfies a renewal-type equation as follows: Proof. By using the ideas of renewal theory and Equation (
2), we obtain
□
Example 1. Suppose that , . Then, we can obtain (see 3.767 in Gradshteyn and Ryzhik (2007))In this case, Equation (
3)
is as follows: Taking into account that , then Equation (
4)
has solution of the form Hence, after calculating the inverse Fourier transform of (Gradshteyn and Ryzhik (2007)), we obtain the density function corresponding toTherefore, the PDF is not dependent on the PDF , and we have the so-called stationary distribution. We remark that even in the case where the Cox process
is Poisson with rate
and
constant, the process
is not the well-known Goldstein–Kac telegraph process, since the particle may or may not change its direction of velocity at renewal instants. Nevertheless, the Goldstein–Kac-type differential equation for
is as follows (
Pogorui et al. (
2021b)):
Since the particle starts its motion from
, we have the following initial conditions:
. From Equation (
3), it follows that
. Hence,
.
If we have a Cox switching process with distribution (
1), it follows from Equation (
5) that the Goldstein–Kac-type differential equation for
(where
is the distribution density of the coordinate of the particle assuming
) is the following:
Theorem 2. Suppose the rate has the PDF and a sequence as are such thatSupposeIf , then satisfies the following diffusion equation: Proof. By using the Cauchy–Schwarz inequality, we have
as .
Therefore,
and by passing to
in the equation
we obtain Equation (
6). □
Corollary 1. According to Theorem 2, if then the telegraph–Cox-based process weakly converges to where is a standard Wiener process. It means that under the conditions of Theorem 2, we have the convergence of the corresponding generators and, hence, the convergence of finite-dimensional distributions of the Cox-based process to .
Example 2. Suppose Then, , . Therefore,
If as such that as , we have Example 3. Let be the PDF of the gamma distribution as follows: Then, and . If as , hence,and Theorem 2 can be applied. That is, if and as such that as , then we have . 4. Telegraph Motion on an Ellipse: Elliptical Telegraph Process
Now, we will investigate the telegraph process
on an ellipse centered at the origin
i.e., a random motion of an object or particle with constant absolute velocity and a switching Cox process governing the direction of movements on the ellipse.
The stochastic process
can be represented as
where
is defined in Equation (
2). By using characteristic functions, we have the expected value
Let us consider some examples:
In the case where
is the Poisson with parameter
and
constant, the characteristic function is of the following form (
Pogorui et al. (
2021c)):
Therefore,
It is easy to see that using Newton’s binomial theorem, we can calculate moments
for any integer
n.
As it was shown above, in the case where
,
, we have
. Hence,
A particle’s motion governed by a telegraph process on a circle (
) was studied extensively in
De Gregorio and Iafrate (
2021), where the authors presented many interesting results. Now, we will develop a partial differential equation that models the motion of a particle on an ellipse governed by a telegraph–Cox stochastic process. Under Kac’s condition, we also obtain a corresponding diffusion equation.
We will call the telegraph–Cox process on an ellipse to with vector representation: .
Consider the following function:
It is not hard to verify that if
, then
and if
, then
If
is the Poisson process
with a rate
, then the two-variate process
is a Markov process with the generative operator
A as follows (
Pogorui et al. (
2021b)):
Let us consider
,
the density function of the particle’s position assuming
. Then, we have the following Kolmogorov equation:
In matrix form, this equation can be written as
It is straightforward to see that is the probability density function of the particle in .
We know that
satisfies the following equation (
Pogorui et al. (
2021b)):
or in an equivalent form as
Under Kac’s condition,
then from Equation (
11), it follows the equation, which can be considered as the diffusion equation on the ellipse,
Applications of Elliptical Telegraph Process
Let us consider the following elliptical telegraph processes:
where
and
is a telegraph process.
Then
converges weakly, as
to Brownian motion on ellipse,
(or elliptical Brownian motion):
where
is a standard Brownian motion. The elliptical Brownian motion can be also presented in a vector form:
Here,
and
We mention that
and
represent the projections of the elliptical telegraph process
on the
x-axis and
y-axis, respectively. Thus, the interpretation can be as a randomized version of an elliptical harmonic oscillator.
Using its formula, we can find the following SDE for
or
Therefore, elliptical Brownian motion can be described by those two SDEs.
The wrapped path of
on
Figure 6 is displayed as the size of the ellipse increases in order to visually distinguish between overlapping arc pieces.