Modern Stochastics: Theory and Applications

The Cox–Ingersoll–Ross (CIR) process, that was first introduced and studied by Cox, Ingersoll and Ross in papers [4–6], can be defined as the process $X=\{{X_{t}},t\ge 0\}$ that satisfies the stochastic differential equation of the form where $W=\{{W_{t}},t\ge 0\}$ is the Wiener process.

The CIR process was originally proposed as a model for interest rate evolution in time and in this framework the parameters a, b and σ have the interpretation as follows: b is the “mean” that characterizes the level, around which the trajectories of X will evolve in a long run; a is “the speed of adjustment” that measures the velocity of regrouping around b; σ is an instantaneous volatility which shows the amplitude of randomness entering the system. Another common application is the stochastic volatility modeling in the Heston model which was proposed in [10] (an extensive bibliography on the subject can be found in [11] and [12]).

For the sake of simplicity, we shall use another parametrization of the equation (1), namely

According to [9], if the condition $2k\ge {\sigma ^{2}}$, sometimes referred to as the Feller condition, holds, then the CIR process is strictly positive. It is also well-known that this process is ergodic and has a stationary distribution.

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 [1, 2, 7, 21]). Therefore, it is reasonable to introduce a fractional Cox–Ingersoll–Ross process, i.e. to modify the equation $\text{(2)}$ by replacing (in some sense) the standard Wiener process with the fractional Brownian motion ${B^{H}}=\{{B_{t}^{H}},t\ge 0\}$, i.e. with a centered Gaussian process with the covariance function $\mathbb{E}[{B_{t}^{H}}{B_{s}^{H}}]=\hspace{2.5pt}\frac{1}{2}({t^{2H}}+{s^{2H}}-|t-s{|^{2H}})$.

There are several approaches to the definition of the fractional Cox–Ingersoll–Ross process. The paper [15] introduces the so-called “rough-path” approach; in [13, 14] it is defined as a time-changed standard CIR process with inverse stable subordinator; another definition is presented in [8] as a part of discussion on rough Heston models.

A simpler pathwise approach is presented in [17] and [18]. There, the fractional Cox–Ingersoll–Ross process was defined as the square of the solution of the SDE until the first moment of zero hitting and zero after this moment (the latter condition was necessary as in the case of $k>0$ the existence of the solution of (3) could not be guaranteed after the first moment of reaching zero).

The reason of such definition lies in the fact that the fractional Cox–Ingersoll–Ross process X defined in that way satisfies pathwisely (until the first moment of zero hitting) the equation where the integral with respect to the fractional Brownian motion is considered as the pathwise Stratonovich integral.

It was shown that if $k>0$ and $H>\frac{1}{2}$, such process is strictly positive and never hits zero; if $k>0$ and $H<\frac{1}{2}$, the probability that there is no zero hitting on the fixed interval $[0,T]$, $T>0$, tends to 1 as $k\to \infty $.

The special case of $k=0$ was considered in [17]. In this situation Y is the well-known fractional Ornstein–Uhlenbeck process (for more details see [3]) and, if $a\ge 0$, the fractional Cox–Ingersoll–Ross process hits zero almost surely, and if $a<0$ the probability of hitting zero is greater than 0 but less than 1.

However, such a definition has a significant disadvantage: according to it, the process remains on the zero level after reaching the latter, and if $H<1/2$, such case cannot be excluded. In this paper we generalize the approach presented in [18] and [17] in order to solve this issue.

We define the fractional CIR process on ${\mathbb{R}_{+}}$ for $H<1/2$ as the square of Y which is the pointwise limit as $\varepsilon \downarrow 0$ of the processes ${Y_{\varepsilon }}$ that satisfy the SDE of the following type: where a, k, $\sigma >0$.

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, Y is continuous and satisfies the equation of the type for all $t\in [\alpha ,\beta ]$ where $(\alpha ,\beta )$ is any interval of Y’s positiveness.

The possibility of getting the equation of the form (3) is restricted due to the obscure structure of the set $\{t\ge 0\hspace{2.5pt}|\hspace{2.5pt}Y(t)=0\}$ which is connected to the structure of the level sets of the fractional Brownian motion. For some results on the latter see, for example, [19].

This paper is organised as follows.

In Section 1 we give preliminary remarks on some results concerning the existence and uniqueness of solutions of SDEs driven by an additive fractional Brownian motion, as well as explain connection of this paper to [17] and [18].

In Section 2 we construct the square root process as the limit of approximations. In particular, it contains a variant of the comparison Lemma and a uniform boundary for all moments of the prelimit processes.

In Section 3 we consider properties of the square root process. We prove that Y is nonnegative and positive a.e., and also continuous.

In Section 4 we give some remarks concerning the equation for the limit square root process on ${\mathbb{R}_{+}}$. We obtain the equation for this process with the noise in form of sum of increments of the fractional Brownian motion on the intervals of Y’s positiveness.

Section 5 is fully devoted to the pathwise definition and equation of the fractional Cox–Ingersoll–Ross process. We prove that on each interval of positiveness the process satisfies the CIR SDE with the integral with respect to a fractional Brownian motion, considered as the pathwise Stratonovich integral. The equation on an arbitrary finite interval is also obtained, with the integral with respect to a fractional Brownian motion considered as the sum of pathwise Stratonovich integrals on intervals of positiveness.

Let ${B^{H}}=\{{B^{H}}(t),t\ge 0\}$ be a fractional Brownian motion with an arbitrary Hurst index $H\in (0,1)$. Consider the process $Y=\{Y(t),\hspace{2.5pt}t\ge 0\}$, such that

Note that, according to [20], the sufficient conditions that guarantee the existence and the uniqueness of the strong solution of the equation are as follows:

(i) for $H<1/2$: linear growth condition: where C is a positive constant;

(ii) for $H>1/2$: Hölder continuity: where C is a positive constant, ${z_{1}},{z_{2}}\in \mathbb{R}$, ${t_{1}},{t_{2}}\in [0,T]$, $1>\alpha >1-1/2H$, $\gamma >H-\frac{1}{2}$.

The function satisfies both (i) and (ii) for all $y\in (\delta ,+\infty )$, where $\delta \in (0,{Y_{0}})$ is arbitrary. Therefore for all $H\in (0,1)$ the unique strong solution of (4) on $[0,T]$ exists until the first moment of hitting the level $\delta \in (0,{Y_{0}})$ and, from the fact that δ is arbitrary, it exists until the first moment of zero hitting.

Let $\tau :=\sup \{t>0\hspace{2.5pt}|\hspace{2.5pt}\forall s\in [0,t):Y(s)>0\}$ be the first moment of zero hitting by Y. The fractional Cox-Ingersoll-Ross process was defined in [18] as follows.

It is known ([18], Theorem 2) that if $H>\frac{1}{2}$, the process Y is strictly positive a.s., therefore the fractional Cox–Ingersoll–Ross process is simply ${Y^{2}}(t)$, $t\ge 0$.

However, in the case of $H<\frac{1}{2}$, the process Y may hit zero. In such situation, according to (7), the corresponding trajectories of the fractional Cox–Ingersoll–Ross process remain in zero after this moment, which is an undesirable property for any financial applications.

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

Consider the process ${Y_{\varepsilon }}=\{{Y_{\varepsilon }}(t),t\in [0,T]\}$ that satisfies the equation of the form where ${Y_{0}},k,a,\sigma >0$ and ${B^{H}}=\{{B^{H}}(t),t\ge 0\}$ is a fractional Brownian motion with $H\in (0,1/2)$.

For any $T>0$, the function ${f_{\varepsilon }}$: $\mathbb{R}\to \mathbb{R}$ such that satisfies the conditions (5) and (6), therefore the strong solution of (8) exists and is unique.

The goal of this section is to prove that there is a pointwise limit of ${Y_{\varepsilon }}$ as $\varepsilon \to 0$.

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

According to Lemma 2.1, for any ${\varepsilon _{1}}>{\varepsilon _{2}}$ and for all $t\in (0,\infty )$:

Now, let us show that there exists the limit

We will need an auxiliary result, presented in [16].

Let a, k, $\sigma >0$, $H\in (0,1)$.

Now let us prove several properties of both square root process and its ε-approximations.

For the next result, we will require the following well-known property of the fractional Brownian motion (see, for example, [16]).

The set $\{t>0\hspace{2.5pt}|\hspace{2.5pt}Y(t)>0\}$ is open in the natural topology on $\mathbb{R}$ a.s., so it can be a.s. represented as the finite or countable union of non-intersecting intervals, i.e. where $({\alpha _{i}},{\beta _{i}})\cap ({\alpha _{j}},{\beta _{j}})=\varnothing $, $i\ne j$.

Moreover, the set $\{t\ge 0\hspace{2.5pt}|\hspace{2.5pt}Y(t)>0\}$ can be a.s. represented as where ${\alpha _{0}}=0$, ${\beta _{0}}$ is the first moment of zero hitting by the square root process Y, $({\alpha _{i}},{\beta _{i}})\cap ({\alpha _{j}},{\beta _{j}})=\varnothing $, $i\ne j$, and $({\alpha _{i}},{\beta _{i}})\cap [{\alpha _{0}},{\beta _{0}})=\varnothing $, $i\ne 0$.

The equation for $Y(t)$ in case of $t\in [0,{\beta _{0}}]$ has already been obtained in Remark 3.3. In order to get the equation for an arbitrary $t\in {\mathbb{R}_{+}}$, consider the following procedure.

Let $\mathcal{I}$ be the set of all non-intersecting open intervals from representation (25), i.e.

Consider an arbitrary interval $(\alpha ,\beta )\in \mathcal{I}$ and assume that $t\in [\alpha ,\beta ]$. Then, from Theorem 3.2,

Consider all such intervals $({\tilde{\alpha }_{j}},{\tilde{\beta }_{j}})\in \mathcal{I}$, $j=1,\dots ,M$, $M\in \mathbb{N}\cup \{\infty \}$, that ${\tilde{\alpha }_{j}}<\alpha $.

For each $j=1,\dots ,M$, Moreover, Therefore

The question whether Y satisfies the equation of the form (28) on ${\mathbb{R}_{+}}$ remains open due to the obscure structure of its zero set. The simulation studies do not contradict with that either.

Indeed, consider the process $Z=\{Z(t),t\ge 0\}$ that satisfies the SDE where Y is, as usual, the pointwise limit of ${Y_{\varepsilon }}$ and ${Y_{0}}$, k, a, σ are positive constants. By following explicitly the proof of Proposition A.1 in [3] it can be shown that

Now assume that Y satisfies the equation of the form (28) for all $t\ge 0$. Then Y admits the representation of the form (29), i.e. so the paths of Y and paths obtained by direct simulation of the right-hand side of (30) must coincide.

In other words, if we simulate the trajectory of Y, apply transform given in the RHS of (30) to it and receive the trajectory that significantly differs from the initial one, it would be an evidence that Y does not satisfy the equation of the form (28) for all $t\ge 0$.

To simulate the left-hand side of (30), the Euler approximations of the process ${Y_{\varepsilon }}$ with $\varepsilon =0.01$ are used. They are compared to the right-hand side of (30) simulated explicitly using the same ${Y_{\varepsilon }}$. The mesh of the partition is $\Delta t=0.0001$, $T=5$, $H=0.4$, ${Y_{0}}=k=a=\sigma =1$ (see Fig. 2).

As we see, the Euler approximation of Y (of ${Y_{\varepsilon }}$ with small ε, black) indeed almost coincides with its transformation given by the right-hand side of (30) (red), so no contradiction is obtained.

Consider a set of random processes $\{{Y_{\varepsilon }},\varepsilon >0\}$ which satisfy the equations of the form driven by the same fractional Brownian motion with the same parameters k, a, $\sigma >0$ and the same starting point ${Y_{0}}>0$.

Let the process $Y=\{Y(t),t\in [0,T]\}$ be such that for all $t\ge 0$:

Let us show that this definition is indeed a generalization of the original Definition 1.1.

First, we will require the following definition.

Taking into account the results of previous sections and from Theorem 1 in [18], for all $t\in [0,{\beta _{0}}]$:

A similar result holds for all $t\in [{\alpha _{i}},{\beta _{i}}]$, $i\ge 1$.

Finally, similarly to Section 4, the next result follows from Theorem 5.1.

Consider an arbitrary $(\alpha ,\beta )\in \mathcal{I}$, where $\mathcal{I}$ is a set of all open intervals from representation (25) (see Section 4). Let ${\beta _{0}}$ be the first moment of zero hitting by Y and $({\tilde{\alpha }_{j}},{\tilde{\beta }_{j}})\in \mathcal{I}$, $j=1,\dots ,M$, $M\in \mathbb{N}\cup \{\infty \}$, are all such intervals that ${\tilde{\alpha }_{j}}<\alpha $.