In this paper we define the fractional Cox–Ingersoll–Ross process as

The classical Cox–Ingersoll–Ross (CIR) process, which was proposed and studied by Cox, Ingersoll and Ross in [

Here

The CIR process is widely used for short-term interest rate modeling as well as for stochastic volatility modeling in the Heston model [

It is well known that the CIR process is ergodic and has a stationary distribution. Moreover, the distribution of its future values

However, the real financial models are often characterized by the so-called “memory phenomenon” (see [

The definition of the fractional CIR process with

In this paper we introduce a natural generalization of the above model. First, we consider the process

Next, for the case of

The paper is organized as follows. In Section

Consider the process

Let

Before moving to the main result of this section, let us give the definition of the pathwise Stratonovich integral.

Let

Let us fix an

According to (

Consider an arbitrary partition of the interval

Using (

Factoring each summand as the difference of squares, we get:

Expanding the brackets in the last expression, we obtain:

Let the mesh

Note that the left-hand side of (

Thus, the fractional Cox–Ingersoll–Ross process, introduced in Definition

In the case of

The next natural question regarding the fractional CIR process is finiteness of its zero hitting time moment. It is obvious that it coincides with the respective moment of the process

Before formulating the main result of the section let us give a well-known property of trajectories of fractional Brownian motion (see, for example, [

The proof is by contradiction.

Let

For all

Consider

The process

From this and Proposition

It is clear that there exists

For

Let us show that there exists

From the choice of

Choosing the corresponding

The contradiction obtained proves the theorem for

The condition of

Let

This lemma holds for an arbitrary Hurst index

Let

As

Assume that

As

Therefore,

Now it is easy to show that

This means that

Finally, as

Now let us move to the main result of the section.

The proof is by contradiction.

Assume that

Let us consider the case of

According to Lemma

Just like in Theorem

The process

Let

According to Proposition

As

Let us transform (

According to [

Therefore, as

Hence, as

If

Theorems

10000 sample paths of the fractional Cox–Ingersoll–Ross process were simulated on the interval

There were no zero hitting for 10000 trajectories simulated for four cases that satisfy the conditions of Theorem

Case of

Case of

Case of

Case of

Case of

Case of

Case of

Case of

However, the behavior of the fractional Cox–Ingersoll–Ross process is not completely clear for the situation of