**1. Introduction**

Modeling inflation dynamics is a tricky topic, particularly in the Eurozone. The determinants of inflation are multifaced, including interest rates, GDP growth, supply and demand of goods and services, exchange rates, etc. Moreover, inflation is somehow persistent over time, such as the low inflation rates in the recent years. In order to model the empirical pattern of inflation, we need a stochastic model with a mean-reversion property as well as time-dependent increments. Both features are mathematically difficult to design because all basic stochastic processes, such as a standard Brownian motion have time-independent increments and it is not mean-reverting.

We propose a novel approach by utilizing a fractional Brownian motion (fBm) and a Lévy process. Both stochastic concepts are relatively new in economic applications. Yet, recent discoveries about fBm's in mathematics already unravel striking insights to economics and finance, such as the modeling of inflation dynamics. We model inflation dynamics by a stochastic process, *Xt*. Before discussing the mathematical details, we provide a brief summary of the relationship across the different stochastic processes (**Figure 1**).

Each of the three stochastic processes have special properties. Interestingly, the overlap of the three stochastic processes gives a subset of new processes with highly interesting and uncommon properties. In this article, we study the subset of a

derivative to non-integer order, in particular *d*1*=*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92292*

physics, biology, engineering, economics and finance.

1

compute fractional derivatives by the following formula:

*D*1

*x*0 *dx*1*=*<sup>2</sup> <sup>¼</sup> *<sup>d</sup>*1*=*<sup>2</sup>

reader is referred to [5]. For *<sup>m</sup>* <sup>¼</sup> <sup>1</sup>

concludes the chapter.

Brownian motion, *BH*

**3**

as it is of today.

semi-derivative of *<sup>d</sup>*1*=*<sup>2</sup>

of Leibniz with Bernoulli and L'Hospital. Laplace, Fourier, Abel and recently up to Riesz Feller and Mishura [4] contributed to the development of fractional calculus

*Modeling Inflation Dynamics with Fractional Brownian Motions and Lévy Processes*

The interest to fractional calculus has to do with its relationship to dynamics and stochastic processes in general. In the past decade, the field of fractional calculus is growing anew due to new discoveries in mathematics and theoretical physics. The first book on fraction calculus is by [5]. Considerable interest in fractional calculus has been stimulated by the many applications in different fields of sciences, such as

Now, let us compute a fractional derivative of a concrete example: What is the

of a constant. From standard calculus, we know that the derivative of a constant is zero. Yet, the half-derivative is not zero as we will see soon. In general, you can

Γð Þ 1 þ *p* � *m*

where Γ (x) is the Gamma function. Similarly, you can compute the fractional derivatives and fractional integrals by the Riemann-Liouville formula. For simplification, we do not introduce the Riemann-Liouville calculus here. The interested

> 2 � � *<sup>x</sup>*<sup>0</sup>�<sup>1</sup>

The result is perhaps the most remarkable result in this brief discussion of fractional calculus. It cannot be embraced too much and deserves a special place in the hall of fame in fractional calculus. Note, the semi-derivative of a constant is surprisingly dependent on *π* and on the variable x. Indeed, this result is utilized

The chapter is organized as follows: Section 2 studies the modeling with fractional Brownian motions. We introduce the concept by defining a fractional Brownian motion in more detail. Section 3 defines a Lévy process and relates it to a Brownian motion. Finally, in Section 4, we start the simulation exercise. We study the stylized facts of inflation rates in the Eurozone from 1997 to 2020. Subsequently, we specify a stochastic differential equation with a fractional Brownian motion and a Lévy process and run several numerical simulations. Section 5

**2. Inflation modeling with fractional Brownian motion (fBm)**

In this section, we define a "fractional Brownian Motion" (fBm). First of all, a fBm is not a (semi-)martingale. Thus, Ito's calculus does not apply anymore. Consequently, the lack of the martingale property has major implications in stochastic calculus. Indeed, one have to develop – similar to Ito's Lemma – completely new stochastic integration and differentiation rules for fractional Brownian motions. We define an ordinary Brownian motion as a special case of a fractional Brownian motion. Indeed, Mandelbrot and van Ness [6] defined a fractional

*<sup>t</sup>* , as a Brownian motion together with a Hurst-Index, *Hϵ*ð Þ 0, 1 ,

*Dmxp* <sup>¼</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>p</sup>*

<sup>2</sup>*x*<sup>0</sup> <sup>¼</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>0</sup> <sup>Γ</sup> <sup>1</sup> <sup>þ</sup> <sup>0</sup> � <sup>1</sup>

repeatedly in fractional calculus in order to simplify solutions.

*dx*1*<sup>=</sup>*2? This example is a semi-derivative or half-derivative

<sup>2</sup> and *p* ¼ 0, we obtain from Eq. (2)

<sup>2</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffi

, is already in the correspondence

*xp*�*<sup>m</sup>*, (2)

*<sup>π</sup><sup>x</sup>* <sup>p</sup> *:* (3)

**Figure 1.** *Overview and Relation of Stochastic Processes. Source: B Herzog (2020).*

fractional Brownian motion (fBm) and a Brownian motion with drift as a subclass of Lévy processes in general. Furthermore, for the first-time, we combine both types of stochastic processes in one model.

The standard Brownian motion is a Gaussian process with independent and stationary increments. However, a fBm is a Gaussian process but does not have independent increments. Similarly, a Brownian motion with drift is a subset of a Lévy process and a Gaussian process. This group of processes belongs to infinitely divisible distributions. We exhibit the relationships and properties between the different types of stochastic processes in order to model the inflation dynamics of the Eurozone.

Let us start with some preliminaries about stochastic processes in general. One can imagine a stochastic process as a sequence of random variables over time, *t*. Let ð Þ Ω, *F*, *P* be a filtered probability space and *X* ¼ *Xt* f g : *t*>0 be a stochastic process on the probability space. The filtration *F* ¼ *Ft* f g : *t* >0 is an increasing flow of information and *P* is defined as a standard probability measure [1].

Furthermore, we need the idea of a stochastic differential equation (SDE) [2]. A non-linear stochastic differential equation for the inflation process, *Xt*, has the form:

$$dX\_t = a(X,t)dt + b(X,t)dB\_t^H,\tag{1}$$

where *a X*ð Þ , *<sup>t</sup> dt* is called the trend-term and *b X*ð Þ , *<sup>t</sup> dBH <sup>t</sup>* the diffusion-term contingent of a fractional Brownian motion, *dBH <sup>t</sup>* . The details of fractional Brownian motions with different "Hurst-Indices," *Hϵ*ð Þ 0, 1 , will be discussed in more detail in Section 2. However, if we choose *<sup>H</sup>* <sup>¼</sup> <sup>1</sup> 2 , the fBm, *B*<sup>1</sup>*=*<sup>2</sup> *<sup>t</sup>* , turns into an ordinary Brownian Motion discovered by Robert Brown in 1827 [1, 3].

The origin and idea of fractional processes or fractional calculus is likewise of interest in general. Indeed, fractional calculus is a subfield in mathematics, which deals with integrals and derivatives of arbitrary order. Fractional calculus is both an old and new field at the same time. It is an old topic since some issues have been discovered by Leibniz and Euler. In fact, the idea of generalizing the notion of a

*Modeling Inflation Dynamics with Fractional Brownian Motions and Lévy Processes DOI: http://dx.doi.org/10.5772/intechopen.92292*

derivative to non-integer order, in particular *d*1*=*<sup>2</sup> , is already in the correspondence of Leibniz with Bernoulli and L'Hospital. Laplace, Fourier, Abel and recently up to Riesz Feller and Mishura [4] contributed to the development of fractional calculus as it is of today.

The interest to fractional calculus has to do with its relationship to dynamics and stochastic processes in general. In the past decade, the field of fractional calculus is growing anew due to new discoveries in mathematics and theoretical physics. The first book on fraction calculus is by [5]. Considerable interest in fractional calculus has been stimulated by the many applications in different fields of sciences, such as physics, biology, engineering, economics and finance.

Now, let us compute a fractional derivative of a concrete example: What is the semi-derivative of *<sup>d</sup>*1*=*<sup>2</sup> *x*0 *dx*1*=*<sup>2</sup> <sup>¼</sup> *<sup>d</sup>*1*=*<sup>2</sup> 1 *dx*1*<sup>=</sup>*2? This example is a semi-derivative or half-derivative of a constant. From standard calculus, we know that the derivative of a constant is zero. Yet, the half-derivative is not zero as we will see soon. In general, you can compute fractional derivatives by the following formula:

$$D^m \mathfrak{x}^p = \frac{\Gamma(1+p)}{\Gamma(1+p-m)} \mathfrak{x}^{p-m},\tag{2}$$

where Γ (x) is the Gamma function. Similarly, you can compute the fractional derivatives and fractional integrals by the Riemann-Liouville formula. For simplification, we do not introduce the Riemann-Liouville calculus here. The interested reader is referred to [5]. For *<sup>m</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> and *p* ¼ 0, we obtain from Eq. (2)

$$D^{\frac{1}{2}}\mathfrak{x}^{0} = \frac{\Gamma(\mathfrak{1}+\mathfrak{0})}{\Gamma\left(\mathfrak{1}+\mathfrak{0}-\frac{1}{2}\right)}\mathfrak{x}^{0-\frac{1}{2}} = \frac{\mathfrak{1}}{\sqrt{\pi\mathfrak{x}}}.\tag{3}$$

The result is perhaps the most remarkable result in this brief discussion of fractional calculus. It cannot be embraced too much and deserves a special place in the hall of fame in fractional calculus. Note, the semi-derivative of a constant is surprisingly dependent on *π* and on the variable x. Indeed, this result is utilized repeatedly in fractional calculus in order to simplify solutions.

The chapter is organized as follows: Section 2 studies the modeling with fractional Brownian motions. We introduce the concept by defining a fractional Brownian motion in more detail. Section 3 defines a Lévy process and relates it to a Brownian motion. Finally, in Section 4, we start the simulation exercise. We study the stylized facts of inflation rates in the Eurozone from 1997 to 2020. Subsequently, we specify a stochastic differential equation with a fractional Brownian motion and a Lévy process and run several numerical simulations. Section 5 concludes the chapter.

## **2. Inflation modeling with fractional Brownian motion (fBm)**

In this section, we define a "fractional Brownian Motion" (fBm). First of all, a fBm is not a (semi-)martingale. Thus, Ito's calculus does not apply anymore. Consequently, the lack of the martingale property has major implications in stochastic calculus. Indeed, one have to develop – similar to Ito's Lemma – completely new stochastic integration and differentiation rules for fractional Brownian motions.

We define an ordinary Brownian motion as a special case of a fractional Brownian motion. Indeed, Mandelbrot and van Ness [6] defined a fractional Brownian motion, *BH <sup>t</sup>* , as a Brownian motion together with a Hurst-Index, *Hϵ*ð Þ 0, 1 ,

fractional Brownian motion (fBm) and a Brownian motion with drift as a subclass of Lévy processes in general. Furthermore, for the first-time, we combine both

The standard Brownian motion is a Gaussian process with independent and stationary increments. However, a fBm is a Gaussian process but does not have independent increments. Similarly, a Brownian motion with drift is a subset of a Lévy process and a Gaussian process. This group of processes belongs to infinitely divisible distributions. We exhibit the relationships and properties between the different types of stochastic processes in order to model the inflation dynamics of

Let us start with some preliminaries about stochastic processes in general. One can imagine a stochastic process as a sequence of random variables over time, *t*. Let ð Þ Ω, *F*, *P* be a filtered probability space and *X* ¼ *Xt* f g : *t*>0 be a stochastic process on the probability space. The filtration *F* ¼ *Ft* f g : *t* >0 is an increasing flow of

Furthermore, we need the idea of a stochastic differential equation (SDE) [2]. A

*<sup>t</sup>* , (1)

*<sup>t</sup>* . The details of fractional Brownian

*<sup>t</sup>* , turns into an ordinary

*<sup>t</sup>* the diffusion-term con-

non-linear stochastic differential equation for the inflation process, *Xt*, has the

*dXt* <sup>¼</sup> *a X*ð Þ , *<sup>t</sup> dt* <sup>þ</sup> *b X*ð Þ , *<sup>t</sup> dBH*

motions with different "Hurst-Indices," *Hϵ*ð Þ 0, 1 , will be discussed in more detail in

The origin and idea of fractional processes or fractional calculus is likewise of interest in general. Indeed, fractional calculus is a subfield in mathematics, which deals with integrals and derivatives of arbitrary order. Fractional calculus is both an old and new field at the same time. It is an old topic since some issues have been discovered by Leibniz and Euler. In fact, the idea of generalizing the notion of a

, the fBm, *B*<sup>1</sup>*=*<sup>2</sup>

2

information and *P* is defined as a standard probability measure [1].

where *a X*ð Þ , *<sup>t</sup> dt* is called the trend-term and *b X*ð Þ , *<sup>t</sup> dBH*

Brownian Motion discovered by Robert Brown in 1827 [1, 3].

tingent of a fractional Brownian motion, *dBH*

Section 2. However, if we choose *<sup>H</sup>* <sup>¼</sup> <sup>1</sup>

types of stochastic processes in one model.

*Overview and Relation of Stochastic Processes. Source: B Herzog (2020).*

*Linear and Non-Linear Financial Econometrics - Theory and Practice*

the Eurozone.

**Figure 1.**

form:

**2**

in the exponent. The parameter *H* is a moving average of the past increments *dBH t* weighted by the kernel ð Þ *t* � *s H*�1*=*2 . Consequently, fractional Brownian motions have the feature that increments are interdependent. The latter property is known as self-similarity, which displace an invariance of the stochastic process with respect to changes of time scale. Almost all other stochastic processes, such as the ordinary Brownian motion or Lévy process have time-independent increments (at least almost surely). They create the famous class of Markov processes.

There exists an alternative definition of a fractional Brownian motion:

*Modeling Inflation Dynamics with Fractional Brownian Motions and Lévy Processes*

2 *t* <sup>2</sup>*<sup>H</sup>* <sup>þ</sup> *<sup>s</sup>*

 *BH t*

**Proof.** To prove that the covariance for a fractional Brownian motion is correct, we remind the reader that the variance of a fractional Brownian motion is defined as

� �<sup>2</sup> h i <sup>þ</sup> *<sup>B</sup><sup>H</sup>*

A trivial corollary is that if *H* ¼ 1*=*2, we obtain for the covariance *Cov Bt* ð Þ¼ , *Bs* min ½ � *t*, *s* , the result of a standard Brownian motion. Similarly, by trivial computation, you can show that the increments of a fBm have mean zero and variance of

<sup>2</sup>*<sup>H</sup>:* Finally, you can demonstrate that two non-overlapping increments of fractional Brownian motions have the property that they are not independent. In

In summary, a fBm has novel properties following empirical observations in economics, yet different to ordinary stochastic processes. Indeed, a fBm has stationary and interdependent increments. Additionally, a fBm is H-self similar,

The rules of fractional integration and fractional differentiation are more sophisticated than the Ito-stochastic calculus. Details about those rules are in [4]. In the remaining part of this section, we demonstrate the empirical patterns of a fractional Brownian motion for different Hurst-Indices over time (**Figure 2**).

*Simulation of fBm for different Hurst-Index.* H ¼ 0*:*1 *(top panel),* H ¼ 0*:*5 *(middle panel),* H ¼ 0*:*9 *(bottom*

<sup>2</sup>*<sup>H</sup>* � ð Þ *<sup>t</sup>* � *<sup>s</sup>* <sup>2</sup>*<sup>H</sup>* h i*:*

<sup>2</sup>*H*. Note, for *<sup>H</sup>* <sup>¼</sup> <sup>1</sup>*=*2 the variance simplifies to the variance of

*s*

� �<sup>2</sup> h i h i

� �<sup>2</sup> h i � *<sup>B</sup><sup>H</sup>*

*<sup>t</sup> be fractional Brownian*

*<sup>t</sup>* � *<sup>B</sup><sup>H</sup> s*

**Proposition.** *Let the Hurst-Index, H, be* 0< *H* <1*, and BH*

*<sup>t</sup>* , *B<sup>H</sup> s* � � <sup>¼</sup> <sup>1</sup>

ordinary Brownian motion. Thus, the covariance can be rewritten as

2

<sup>2</sup>*<sup>H</sup>* � j j *<sup>t</sup>* � *<sup>s</sup>* <sup>2</sup>*<sup>H</sup>* h i*:*

*motion. The covariance of a fractional Brownian motion is*

*Cov BH*

*<sup>t</sup> BH s* � � <sup>¼</sup> <sup>1</sup>

*Var Bt* � *Bs* ½ �¼ ð Þ *t* � *s*

j j *t* � *s*

**Figure 2.**

**5**

*panel). Source: B Herzog (2020).*

meaning that *BH*

*Cov Bt* ð Þ¼ , *Bs BH*

fact, they are interdependent!

¼ 1 2 *t* <sup>2</sup>*<sup>H</sup>* <sup>þ</sup> *<sup>s</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92292*

*at* <sup>¼</sup> *<sup>a</sup>HBH t* .

Empirically, however, there is evidence that economic and particularly financial time-series have a spectral density with a sharp peak. Additionally, we observe the phenomena of extremely long interdependence of certain trends over time in economics and finance. This presence of interdependence between past increments, directly speaks for the modeling with fractional Brownian motions. A standard Brownian motion is defined by the following properties:

1.*Bt* is almost surely continuous; *Bt*¼<sup>0</sup> ¼ 0;

2.The increments *Bt* � *Bs* for *t*> *s* have mean zero and variance *t* � *s*;

3.The increments *Bt* � *Bs* are independent over time and stationary.

Indeed, we know that the variance of the increment is of *Var Bt* � *Bs* ½ �¼ ð Þ *Bt* � *Bs* <sup>2</sup> h i <sup>¼</sup> *dB*<sup>2</sup> *t* � �. Likewise, the standard deviation is: *<sup>σ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Var Bt* � *Bs* ½ � <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffi *dB*<sup>2</sup> *t* q � *dt*<sup>1</sup>*=*<sup>2</sup> . This is often referred to as the *t* <sup>1</sup>*<sup>=</sup>*2-law. Now, we are ready to define a fractional Brownian motion:

**Definition "Fractional Brownian Motion (fBm)."** *Let the Hurst-Index, H , be* 0< *H* < 1*, then we call B<sup>H</sup> <sup>t</sup> a fractional Brownian Motion with parameter H, such as*

$$\mathcal{B}\_{t=0}^H = \mathcal{B}\_0$$

*and;*

$$B\_t^H - B\_0^H = \frac{\mathbf{1}}{\Gamma\left(H + \frac{1}{2}\right)} \left[ \int\_{-\infty}^0 \left[ (t - s)^{H - \frac{1}{2}} - (-s)^{H - \frac{1}{2}} \right] dB\_t + \int\_0^t (t - s)^{H - \frac{1}{2}} dB\_t \right].$$

Part two of the definition is the so-called Weyl fractional integral. Equivalently, you can use the more intuitive Riemann-Liouville fractional integral, defined by

$$B\_t^H - B\_0^H = \frac{1}{\Gamma\left(H + \frac{1}{2}\right)} \int\_0^t (t - s)^{H - \frac{1}{2}} dB\_s \tag{4}$$

where *<sup>Γ</sup> <sup>H</sup>* <sup>þ</sup> <sup>1</sup> 2 � � is the Gamma function. The rules about fractional integration and fractional differentiation are discussed in detail in [5]. It trivially follows that for *H* ¼ 1*=*2, we obtain the ordinary Brownian Motion, *Bt*. For other values of *H*, such as 0 < *H* < 1*=*2 and 1*=*2< *H* <1 the fractional Brownian Motion *BH <sup>t</sup>* is a fractional derivative or integral. Note, if 0< *H* <1*=*2 we say it has the property of counter persistent or short memory. This is associated with negative correlation. Vice versa for 1*=*2 < *H* < 1, we say it is persistent. This is associated with positive correlation. Thus, modeling with fractional Brownian motions display the property of short- and long-term memory, a property very common in economic and financial time-series.

*Modeling Inflation Dynamics with Fractional Brownian Motions and Lévy Processes DOI: http://dx.doi.org/10.5772/intechopen.92292*

There exists an alternative definition of a fractional Brownian motion:

**Proposition.** *Let the Hurst-Index, H, be* 0< *H* <1*, and BH <sup>t</sup> be fractional Brownian motion. The covariance of a fractional Brownian motion is*

$$Cov\left(B\_t^H, B\_s^H\right) = \frac{1}{2} \left[t^{2H} + s^{2H} - \left(t - s\right)^{2H}\right].$$

**Proof.** To prove that the covariance for a fractional Brownian motion is correct, we remind the reader that the variance of a fractional Brownian motion is defined as *Var Bt* � *Bs* ½ �¼ ð Þ *t* � *s* <sup>2</sup>*H*. Note, for *<sup>H</sup>* <sup>¼</sup> <sup>1</sup>*=*2 the variance simplifies to the variance of ordinary Brownian motion. Thus, the covariance can be rewritten as

$$\begin{split} Cov(B\_t, B\_t) &= \mathbb{E}\left[B\_t^H B\_s^H\right] = \frac{1}{2} \left[ \mathbb{E}\left[ \left(B\_t^H\right)^2 \right] + \mathbb{E}\left[ \left(B\_s^H\right)^2 \right] - \mathbb{E}\left[ \left(B\_t^H - B\_s^H\right)^2 \right] \right] \\ &= \frac{1}{2} \left[ t^{2H} + s^{2H} - \left|t - s\right|^{2H} \right]. \end{split}$$

A trivial corollary is that if *H* ¼ 1*=*2, we obtain for the covariance *Cov Bt* ð Þ¼ , *Bs* min ½ � *t*, *s* , the result of a standard Brownian motion. Similarly, by trivial computation, you can show that the increments of a fBm have mean zero and variance of j j *t* � *s* <sup>2</sup>*<sup>H</sup>:* Finally, you can demonstrate that two non-overlapping increments of fractional Brownian motions have the property that they are not independent. In fact, they are interdependent!

In summary, a fBm has novel properties following empirical observations in economics, yet different to ordinary stochastic processes. Indeed, a fBm has stationary and interdependent increments. Additionally, a fBm is H-self similar, meaning that *BH at* <sup>¼</sup> *<sup>a</sup>HBH t* .

The rules of fractional integration and fractional differentiation are more sophisticated than the Ito-stochastic calculus. Details about those rules are in [4]. In the remaining part of this section, we demonstrate the empirical patterns of a fractional Brownian motion for different Hurst-Indices over time (**Figure 2**).

#### **Figure 2.**

*Simulation of fBm for different Hurst-Index.* H ¼ 0*:*1 *(top panel),* H ¼ 0*:*5 *(middle panel),* H ¼ 0*:*9 *(bottom panel). Source: B Herzog (2020).*

in the exponent. The parameter *H* is a moving average of the past increments *dBH*

have the feature that increments are interdependent. The latter property is known as self-similarity, which displace an invariance of the stochastic process with respect to changes of time scale. Almost all other stochastic processes, such as the ordinary Brownian motion or Lévy process have time-independent increments (at least

Empirically, however, there is evidence that economic and particularly financial time-series have a spectral density with a sharp peak. Additionally, we observe the phenomena of extremely long interdependence of certain trends over time in economics and finance. This presence of interdependence between past increments, directly speaks for the modeling with fractional Brownian motions. A standard

. Consequently, fractional Brownian motions

*H*�1*=*2

*Linear and Non-Linear Financial Econometrics - Theory and Practice*

almost surely). They create the famous class of Markov processes.

2.The increments *Bt* � *Bs* for *t*> *s* have mean zero and variance *t* � *s*;

3.The increments *Bt* � *Bs* are independent over time and stationary.

Indeed, we know that the variance of the increment is of *Var Bt* � *Bs* ½ �¼

*BH <sup>t</sup>*¼<sup>0</sup> ¼ *B*<sup>0</sup>

ð Þ *t* � *s*

<sup>0</sup> <sup>¼</sup> <sup>1</sup> *<sup>Γ</sup> <sup>H</sup>* <sup>þ</sup> <sup>1</sup> 2 � �

such as 0 < *H* < 1*=*2 and 1*=*2< *H* <1 the fractional Brownian Motion *BH*

*<sup>H</sup>*�<sup>1</sup>

and fractional differentiation are discussed in detail in [5]. It trivially follows that for *H* ¼ 1*=*2, we obtain the ordinary Brownian Motion, *Bt*. For other values of *H*,

tional derivative or integral. Note, if 0< *H* <1*=*2 we say it has the property of counter persistent or short memory. This is associated with negative correlation. Vice versa for 1*=*2 < *H* < 1, we say it is persistent. This is associated with positive correlation. Thus, modeling with fractional Brownian motions display the property of short- and long-term memory, a property very common in economic and

h i

Part two of the definition is the so-called Weyl fractional integral. Equivalently, you can use the more intuitive Riemann-Liouville fractional integral, defined by

<sup>2</sup> � �ð Þ*s*

ð*t* 0 ð Þ *t* � *s*

� � is the Gamma function. The rules about fractional integration

**Definition "Fractional Brownian Motion (fBm)."** *Let the Hurst-Index, H , be*

� �. Likewise, the standard deviation is: *<sup>σ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>t</sup> a fractional Brownian Motion with parameter H, such as*

*<sup>H</sup>*�<sup>1</sup> 2

� �

*dBt* þ

*<sup>H</sup>*�<sup>1</sup>

ð*t* 0 ð Þ *t* � *s*

Brownian motion is defined by the following properties:

. This is often referred to as the *t*

ð0 �∞

*BH <sup>t</sup>* � *BH*

1.*Bt* is almost surely continuous; *Bt*¼<sup>0</sup> ¼ 0;

<sup>¼</sup> *dB*<sup>2</sup> *t*

<sup>0</sup> <sup>¼</sup> <sup>1</sup> *<sup>Γ</sup> <sup>H</sup>* <sup>þ</sup> <sup>1</sup> 2 � �

2

weighted by the kernel ð Þ *t* � *s*

ð Þ *Bt* � *Bs* <sup>2</sup> h i

� *dt*<sup>1</sup>*=*<sup>2</sup>

fractional Brownian motion:

0< *H* < 1*, then we call B<sup>H</sup>*

ffiffiffiffiffiffiffi *dB*<sup>2</sup> *t* q

*and;*

*BH <sup>t</sup>* � *<sup>B</sup><sup>H</sup>*

where *<sup>Γ</sup> <sup>H</sup>* <sup>þ</sup> <sup>1</sup>

financial time-series.

**4**

*t*

*Var Bt* � *Bs* ½ � <sup>p</sup> <sup>¼</sup>

*<sup>H</sup>*�<sup>1</sup> <sup>2</sup>*dBt*

<sup>2</sup>*dBs* (4)

*:*

*<sup>t</sup>* is a frac-

<sup>1</sup>*<sup>=</sup>*2-law. Now, we are ready to define a

For H ¼ 0*:*1, we obtain in the top-panel a time-series with short-term memory (**Figure 2**). Contrary in the bottom panel (H ¼ 0*:*9), we observe a strong interdependence or a non-stationary stochastic process. This process reflects longterm memory. The middle panel (H ¼ 0*:*5) denotes a standard Brownian motion. It is interesting that a fractional Brownian motion is a generalization of a standard Brownian motion. **Figure 2** summarizes the different empirical patterns in relationship to the H-Index.

combined with a compound Poisson process. The last process is labeled a jump-

*Modeling Inflation Dynamics with Fractional Brownian Motions and Lévy Processes*

In order to identify Lévy processes, we use the property of infinitely divisible distributions. As soon as you can show that a process belongs to the class of infinitely divisible distributions, you immediately say that this process is a Lévy process. Indeed, there is an intimate relationship of Lévy processes to infinitely

**Definition "Infinitely divisible distribution."** *A real-valued random variable X has an infinitely divisible distribution if for each n* ¼ 1, 2, … *there exist a sequence of independent, identical distributed random variables X*1,*n, X*2,*n*, … *Xn*,*n, such that*

*X* ≔ *X*1,*<sup>n</sup>* þ *X*2,*<sup>n</sup>* þ … þ *Xn*,*<sup>n</sup>*

One way to establish whether a given random variable has an infinitely divisible distribution is via the study of the exponent of the characteristic function. This idea is summarized by the rather sophisticated concept of the Lévy-Khintchine formula

In this subsection, we briefly show that a Brownian motion is a Lévy process. Suppose a Gaussian random variable with distribution *<sup>X</sup>* � *<sup>N</sup> <sup>μ</sup>*, *<sup>σ</sup>*<sup>2</sup> ð Þ and the char-

motion follow a Gaussian process. By the characteristic function, we show that the increments of the Brownian motion are stationary and independent. Thus it strat-

*ϕXt*þ*<sup>s</sup>* ¼ *ϕXt* ∗ *ϕXs*

*<sup>i</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>X</sup><sup>n</sup>*

divisible by *n* and it consists of independent, identical distributed (i.i.d) increments.

**Remark.** Markov processes are the best-known family of stochastic processes in mathematical probability theory. Informally, a Markov process has the property that the future behavior of the process depends on the past only. One can show that Lévy processes are related to Markov processes and even simplify the theory significantly. The link between both stochastic processes is so-called random-stopping times. One can show that a random-stopping time on a Lévy process has the Markov property. Consequently, Lévy processes concern many aspects of probability theory and its applications.

In this section, we simulate different fractional Brownian motions and Lévy processes. The simulation reveals different new patterns of inflation dynamics. Our

*<sup>n</sup>* , *<sup>σ</sup>*<sup>2</sup> *n*

Consequently, a Brownian motion satisfies the properties of a Lévy process.

Eq. (5) demonstrates that the Brownian motion is an infinitely divisible distribution. Eq. (6) shows that the Brownian motion has independent and stationary increments. Thus, we find that the random variable *X* is Lévy by computing the sum of n-

*<sup>n</sup>* with each *X<sup>n</sup>*

*ϕn Xt* <sup>¼</sup> *<sup>e</sup> <sup>i</sup><sup>μ</sup> n*�<sup>1</sup> 2 *t* 2*σ*2 *n <sup>n</sup>*

. We know that the increments of a Brownian

*:* (6)

*<sup>i</sup>* � *<sup>N</sup> <sup>μ</sup>*

. Hence, the Brownian motion is infinitely

*<sup>n</sup>* , *<sup>σ</sup>*<sup>2</sup> *n*

. Therefore,

(5)

*the process X has the same distribution as the processes of X*1,*n, X*2,*n*, … *Xn*,*n.*

process because it exhibits random jumps.

*DOI: http://dx.doi.org/10.5772/intechopen.92292*

**3.2 A Brownian motion is a Lévy process**

ðÞ¼ *<sup>t</sup> <sup>e</sup><sup>i</sup>μ*�<sup>1</sup> 2 *t* <sup>2</sup>*σ*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>X</sup><sup>n</sup>*

<sup>1</sup> � *<sup>N</sup> <sup>μ</sup>*

divisible distributions in general.

(e.g. in [7]).

acteristic function of *ϕXt*

random variables *<sup>X</sup>* <sup>¼</sup> *<sup>X</sup><sup>n</sup>*

we obtain *<sup>X</sup>* � *<sup>N</sup> <sup>μ</sup>*, *<sup>σ</sup>*<sup>2</sup> ð Þ and *<sup>X</sup><sup>n</sup>*

**4. Numerical simulation**

**7**

ifies the Lévy process properties:
