In this paper the fractional Cox–Ingersoll–Ross process on

The Cox–Ingersoll–Ross (CIR) process, that was first introduced and studied by Cox, Ingersoll and Ross in papers [

The CIR process was originally proposed as a model for interest rate evolution in time and in this framework the parameters

For the sake of simplicity, we shall use another parametrization of the equation (

According to [

However, in reality the dynamics on financial markets is characterized by the so-called “memory phenomenon”, which cannot be reflected by the standard CIR model (for more details on financial markets with memory, see [

There are several approaches to the definition of the fractional Cox–Ingersoll–Ross process. The paper [

A simpler pathwise approach is presented in [

The reason of such definition lies in the fact that the fractional Cox–Ingersoll–Ross process

It was shown that if

The special case of

However, such a definition has a significant disadvantage: according to it, the process remains on the zero level after reaching the latter, and if

We define the fractional CIR process on

We prove that this limit indeed exists, is nonnegative a.s. and is positive a.e. with respect to the Lebesgue measure a.s. Moreover,

The possibility of getting the equation of the form (

This paper is organised as follows.

In Section

In Section

In Section

In Section

Section

Let

Note that, according to [

(i) for

(ii) for

The function

Let

The fractional Cox–Ingersoll–Ross (CIR) process is the process

Further,

It is known ([

However, in the case of

Our goal is to modify the definition of the fractional CIR process in order to remove the problem mentioned above.

Consider the process

We will sometimes call

For any

The goal of this section is to prove that there is a pointwise limit of

First, let us prove the analogue of the comparison Lemma.

Let

Denote

The function

It is clear that

Assume that

As

Therefore,

It is obvious that Lemma

Moreover, the condition (i) can be replaced by

According to Lemma

Now, let us show that there exists the limit

We will need an auxiliary result, presented in [

Let

Let an arbitrary

Let us prove that for all

In order to make the further proof more convenient for the reader, we shall divide it into 3 steps. In Steps 1 and 2, we will separately show that (

If

For all

Note that the inequality (

Denote

In addition, if

Furthermore, from Jensen’s inequality,

Finally, from (

Therefore, (

The proof immediately follows from Lemma

Let

From Lemma

The upper bound for

First, note that the trajectories of

Let

For all

It follows immediately from monotonicity of

Later it will be shown that

We will sometimes refer to the limit process

Further, we will consider only finite and integrable paths of

Now let us prove several properties of both square root process and its

Let

From the definition of

Assume that there exists a sequence

In such case, as

According to the Fatou lemma,

For the next result, we will require the following well-known property of the fractional Brownian motion (see, for example, [

Let an arbitrary

Let an arbitrary

It is clear that for all

Indeed,

It is easy to check that the second derivative of the right-hand side of (

It is clear that for all

Indeed, denote

It is known that

Next, denote

Therefore, similarly to (

We shall divide the proof into 3 steps.

Hence for all

It is enough to prove that for any

Indeed, let us fix an arbitrary

Denote

From the continuity with respect to

It is obvious that

It is obvious that

Moreover, for all

Now consider an arbitrary

Denote

However, as for all

From the continuity of

Therefore

Hence, as

So, for all

Hence, for each all

This equation has a unique continuous solution, therefore

From Theorem

The claim follows directly from Theorem

The set

Moreover, the set

Let

1) Proofs for both left and right ends of the segment are similar, so we shall give a proof for the left end only.

Assume that

Let

From continuity of

Therefore

2) From Theorem

Hence, for all

The right-hand side of (

Due to Lemma

To get the result for

Similarly to Theorem

The choice of

By following the proofs of the results above, it can be verified that all of them hold for the resulting limit process

Indeed, let

As

Assume that there is such

Note that, due to continuity of

However, as

The obtained contradiction shows that for all

Simulations illustrate that the processes indeed coincide (see Fig.

Comparison of

The equation for

Let

Consider an arbitrary interval

Consider all such intervals

For each

The question whether

Indeed, consider the process

Now assume that

In other words, if we simulate the trajectory of

To simulate the left-hand side of (

Comparison of right-hand and left-hand sides of (

As we see, the Euler approximation of

Consider a set of random processes

Let the process

The fractional Cox–Ingersoll–Ross process is the process

Let us show that this definition is indeed a generalization of the original Definition

First, we will require the following definition.

Let

Taking into account the results of previous sections and from Theorem 1 in [

A similar result holds for all

We shall follow the proof of Theorem 1 from [

Let us fix an

It is clear that

Consider an arbitrary partition of the interval

Let the mesh

Note that the left-hand side of (

Thus, the process

Finally, similarly to Section

Consider an arbitrary

We are deeply grateful to anonymous referees whose valuable comments helped us to improve the manuscript significantly.