**3. Inflation modeling with Lévy processes**

On first encounter, a Poisson process and a Brownian motion seem to be considerably different. Firstly, a Brownian motion has continuous paths whereas a Poisson process does not. Secondly, a Poisson process is a non-decreasing process and thus has paths of bounded variation over finite time horizons, whereas a Brownian motion does not have monotone paths. In fact, the Brownian motion has unbounded variation over finite time horizons.

Yet, both stochastic processes have a lot in common. Both processes are right continuous with left limits (so-called càdlàg). Consequently, we use these common properties to define a general class of stochastic processes, which are so-called Lévy processes. The class of Lévy processes is rather rich, and the Brownian motion or Poisson process are two prominent subcases.

In general, Lévy processes play a major role in several fields of sciences, such as physics, engineering, economics and mathematical finance. Lévy processes are becoming fashionable to describe the observed reality of financial markets more accurately than models based on a Brownian motion alone. Lévy processes result in a more realistic modeling because it captures the empirical reality of jumpdiffusions. Indeed, asset prices have jumps and spikes and thus risk managers have to consider Lévy processes in order to hedge the risks appropriately. Similarly, the pattern of implied volatility or incomplete markets is reliant to Lévy processes too.

#### **3.1 Introduction to Lévy processes**

The term Lévy process honors the work of the French mathematician Paul Lévy in the 1940s. He pioneered the understanding and characterization of stochastic processes with stationary and independent increments.

**Definition 'Lévy Process."** *A process X* ¼ *Xt* f g : *t*>0 *defined on a probability space* ð Þ *Ω*, *F*, *P is said to be a Lévy process if it possesses the following properties:*

$$P(X\_0 = \mathbf{0}) = \mathbf{1}.$$


The definition does not immediately make visible the richness of the class of Lévy processes. One simple Lévy process is a Brownian motion with drift. Other examples of Lévy processes are the Poisson process. Or a Brownian motion

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

combined with a compound Poisson process. The last process is labeled a jumpprocess because it exhibits random jumps.

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 divisible distributions in general.

**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*

$$X \coloneqq X\_{1,n} + X\_{2,n} + \dots + X\_{n,n}$$

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

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 (e.g. in [7]).

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

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 characteristic function of *ϕXt* ðÞ¼ *<sup>t</sup> <sup>e</sup><sup>i</sup>μ*�<sup>1</sup> 2 *t* <sup>2</sup>*σ*<sup>2</sup> . We know that the increments of a Brownian motion follow a Gaussian process. By the characteristic function, we show that the increments of the Brownian motion are stationary and independent. Thus it stratifies the Lévy process properties:

$$\phi\_{X\_t}^n = e^{\left(\frac{i\mu}{n} - \frac{\mu^2 \sigma^2}{2}\right)^n} \tag{5}$$

$$
\phi\_{X\_{t+s}} = \phi\_{X\_t} \* \phi\_{X\_t}.\tag{6}
$$

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 nrandom variables *<sup>X</sup>* <sup>¼</sup> *<sup>X</sup><sup>n</sup>* <sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>X</sup><sup>n</sup> <sup>i</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>X</sup><sup>n</sup> <sup>n</sup>* with each *X<sup>n</sup> <sup>i</sup>* � *<sup>N</sup> <sup>μ</sup> <sup>n</sup>* , *<sup>σ</sup>*<sup>2</sup> *n* . Therefore, 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>* <sup>1</sup> � *<sup>N</sup> <sup>μ</sup> <sup>n</sup>* , *<sup>σ</sup>*<sup>2</sup> *n* . Hence, the Brownian motion is infinitely divisible by *n* and it consists of independent, identical distributed (i.i.d) increments. Consequently, a Brownian motion satisfies the properties of a Lévy process.

**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.

## **4. Numerical simulation**

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

For H ¼ 0*:*1, we obtain in the top-panel a time-series with short-term memory

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 rela-

On first encounter, a Poisson process and a Brownian motion seem to be considerably different. Firstly, a Brownian motion has continuous paths whereas a Poisson process does not. Secondly, a Poisson process is a non-decreasing process and thus has paths of bounded variation over finite time horizons, whereas a Brownian motion does not have monotone paths. In fact, the Brownian motion has

Yet, both stochastic processes have a lot in common. Both processes are right continuous with left limits (so-called càdlàg). Consequently, we use these common properties to define a general class of stochastic processes, which are so-called Lévy processes. The class of Lévy processes is rather rich, and the Brownian motion or

In general, Lévy processes play a major role in several fields of sciences, such as

physics, engineering, economics and mathematical finance. Lévy processes are becoming fashionable to describe the observed reality of financial markets more accurately than models based on a Brownian motion alone. Lévy processes result in

diffusions. Indeed, asset prices have jumps and spikes and thus risk managers have to consider Lévy processes in order to hedge the risks appropriately. Similarly, the pattern of implied volatility or incomplete markets is reliant to Lévy processes too.

The term Lévy process honors the work of the French mathematician Paul Lévy in the 1940s. He pioneered the understanding and characterization of stochastic

**Definition 'Lévy Process."** *A process X* ¼ *Xt* f g : *t*>0 *defined on a probability*

*P X*ð Þ¼ <sup>0</sup> ¼ 0 1*:*

Mathematically, *X* is stochastically continuous for every 0 <*t*<*T* and *ε* >0

2.For 0 <*s*<*t*, the increments *Xt* � *Xs* are stationary and equal in distribution to *Xt*�*<sup>s</sup>*, i.e. the increment have the same distribution whenever time elapses.

3.For 0< *s*<*t*, the increment *Xt* � *Xs* is independent of f g *Xu* : *u*>*s* or we say the

The definition does not immediately make visible the richness of the class of Lévy processes. One simple Lévy process is a Brownian motion with drift. Other examples of Lévy processes are the Poisson process. Or a Brownian motion

*space* ð Þ *Ω*, *F*, *P is said to be a Lévy process if it possesses the following properties:*

1.The paths of *X* are *P*-almost surely right continuous with left limits.

a more realistic modeling because it captures the empirical reality of jump-

(**Figure 2**). Contrary in the bottom panel (H ¼ 0*:*9), we observe a strong

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

tionship to the H-Index.

**3. Inflation modeling with Lévy processes**

unbounded variation over finite time horizons.

Poisson process are two prominent subcases.

**3.1 Introduction to Lévy processes**

such as *log <sup>s</sup>*!*<sup>t</sup>*

**6**

processes with stationary and independent increments.

*P X*ð Þ¼ *<sup>t</sup>* � *Xs* >*ε* 0.

increment is independent of filtration *Fs*.

model is calibrated to the monthly frequency of the past inflation dynamics in the Eurozone from 1997 to 2020.

The simulation follows a mean-reverting stochastic differential equation driven by a fractional Brownian motion and a Lévy process. Suppose *Xt* denotes the inflation process over time *t*. We model the inflation dynamics by a stochastic differential equation of the form

$$dX\_t = (\alpha - \beta \* X\_t)dt + \sigma dB\_t^H + N(\mu, \gamma)dN(\lambda) \tag{7}$$

where *α* and *β* are the mean-reversion trends and *σ* denotes the volatility coming from the fractional Brownian motion, *B<sup>H</sup> <sup>t</sup>* . The parameter *H* reflects the Hurst-Index of the fractional Brownian motion. The last term is a jump-process modelled by a Poisson process, *N*ð Þ *μ*, *γ* , with parameters *μ* and *γ*. The jump-frequency is of *λ*.

The numerical simulation is computed over 1000 time steps and over 1000 different stochastic processes. The Eurozone inflation data are downloaded from the ECB Statistical Data Warehouse. We calibrate the model to the aggregate inflation dynamics of the Eurozone (**Figures 3** and **4**).

**Figure 3** represents the Harmonized Index of Consumer Price (HICP) of the Eurozone on monthly frequency from 1997 to 2020. One clearly sees the sharp drop in inflation rates during the global financial crisis of 2008–2009. Subsequently inflation rebounded, however, afterwards with low inflation rates, partly deflation, in the years of 2013–2016. In recent years, inflation rates were in the range of 1.0– 2.0%. Thus, the inflation rate in the Eurozone is following Article 127 TFEU and the definition of price-stability by the European Central Bank [8]: " … inflation rates below, but close to 2% over the medium term."

Based on the inflation data, we compute the histogram of Eurozone inflation rates in **Figure 4**. The distribution displays particularly a right-skewedness. Indeed, the mean is of 1.66, the median of 1.80 and the modus is of 2.10. Moreover, the standard deviation is of 0.77, the variance of 0.60, the skewness of �0.22 and the kurtosis of �0.06 is almost zero. These parameters characterize the Eurozone's inflation rate properties over time.

Next, we choose the following parameters in our stochastic differential equation (Eq. (7)): *α* ¼ 1*:*7, *β* ¼ 1*:*0, *σ* ¼ 0*:*4, *μ* ¼ �2*:*0, *γ* ¼ 0*:*5, *λ* ¼ 0*:*01 and *H* ¼ 0*:*2. We run the simulation model for 1000-time steps. **Figure 5** represents the result of one simulation, where the mean is of 1.60, the median of 1.73, the variance of 0.81 and

> the skewness of �0.42. This demonstrates that the simulation is following the distribution properties of inflation data, particularly the right-skewedness.

*Simulation of Inflation Dynamics according to equation (7). Top panel denotes the inflation rate and bottom*

*Histogram of Eurozone Inflation Rates. Data from ECB Data Warehouse. Source: B Herzog (2020).*

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

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

Eurozone.

**9**

**Figure 5.**

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

**Figure 4.**

It turns out that the simulation replicates the distributional properties quite well, except for the kurtosis. Nonetheless, we clearly see in the bottom panel of **Figure 5** that the distribution is right-skewed with more tail events on the left-hand side. If we run the same model with the Gaussian assumption, by using a standard Brownian motion, *H* ¼ 0*:*5, we obtain a somewhat different result. The mean is of 0.94, the median of 0.85, the variance of 1.23, the skewness of 0.45 and the kurtosis of 2.64. This distribution is not right-skewed and has higher variance than the stylized facts. Hence, we conclude that a fractional Brownian motion with a Lévy process provide a better approach in order to model the inflation dynamics of the

Finally, we discuss the results of the simulation exercise with 1000 runs. In this simulation, we have specified our stochastic differential equation (Eq. (7)) as follows: *α* ¼ 1*:*7, *β* ¼ 0, *σ* ¼ 0*:*3, *μ* ¼ �2*:*0, *γ* ¼ 0*:*1, *λ* ¼ 0*:*00 and *H* ¼ 0*:*2. **Figure 6** represents in the top-panel the stochastic paths of all stochastic processes and in the bottom-panel the respective histogram. The numerical simulation yields a mean and median of approximately 1.7, a variance of 1.4 and negative skewness of �1.71.

**Figure 3.** *Eurozone HICP-Inflation Rate. Data from ECB Data Warehouse. Source: B Herzog (2020).*

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

**Figure 4.**

model is calibrated to the monthly frequency of the past inflation dynamics in the

by a fractional Brownian motion and a Lévy process. Suppose *Xt* denotes the inflation process over time *t*. We model the inflation dynamics by a stochastic

*dXt* <sup>¼</sup> ð Þ *<sup>α</sup>* � *<sup>β</sup>* <sup>∗</sup>*Xt dt* <sup>þ</sup> *<sup>σ</sup>dB<sup>H</sup>*

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

The simulation follows a mean-reverting stochastic differential equation driven

where *α* and *β* are the mean-reversion trends and *σ* denotes the volatility coming

of the fractional Brownian motion. The last term is a jump-process modelled by a Poisson process, *N*ð Þ *μ*, *γ* , with parameters *μ* and *γ*. The jump-frequency is of *λ*. The numerical simulation is computed over 1000 time steps and over 1000 different stochastic processes. The Eurozone inflation data are downloaded from the ECB Statistical Data Warehouse. We calibrate the model to the aggregate infla-

**Figure 3** represents the Harmonized Index of Consumer Price (HICP) of the Eurozone on monthly frequency from 1997 to 2020. One clearly sees the sharp drop in inflation rates during the global financial crisis of 2008–2009. Subsequently inflation rebounded, however, afterwards with low inflation rates, partly deflation, in the years of 2013–2016. In recent years, inflation rates were in the range of 1.0– 2.0%. Thus, the inflation rate in the Eurozone is following Article 127 TFEU and the definition of price-stability by the European Central Bank [8]: " … inflation rates

Based on the inflation data, we compute the histogram of Eurozone inflation rates in **Figure 4**. The distribution displays particularly a right-skewedness. Indeed, the mean is of 1.66, the median of 1.80 and the modus is of 2.10. Moreover, the standard deviation is of 0.77, the variance of 0.60, the skewness of �0.22 and the kurtosis of �0.06 is almost zero. These parameters characterize the Eurozone's

Next, we choose the following parameters in our stochastic differential equation (Eq. (7)): *α* ¼ 1*:*7, *β* ¼ 1*:*0, *σ* ¼ 0*:*4, *μ* ¼ �2*:*0, *γ* ¼ 0*:*5, *λ* ¼ 0*:*01 and *H* ¼ 0*:*2. We run the simulation model for 1000-time steps. **Figure 5** represents the result of one simulation, where the mean is of 1.60, the median of 1.73, the variance of 0.81 and

*Eurozone HICP-Inflation Rate. Data from ECB Data Warehouse. Source: B Herzog (2020).*

*<sup>t</sup>* þ *N*ð Þ *μ*, *γ dN*ð Þ*λ* (7)

*<sup>t</sup>* . The parameter *H* reflects the Hurst-Index

Eurozone from 1997 to 2020.

differential equation of the form

from the fractional Brownian motion, *B<sup>H</sup>*

tion dynamics of the Eurozone (**Figures 3** and **4**).

below, but close to 2% over the medium term."

inflation rate properties over time.

**Figure 3.**

**8**

#### **Figure 5.**

*Simulation of Inflation Dynamics according to equation (7). Top panel denotes the inflation rate and bottom panel the histogram. Source: B Herzog (2020).*

the skewness of �0.42. This demonstrates that the simulation is following the distribution properties of inflation data, particularly the right-skewedness.

It turns out that the simulation replicates the distributional properties quite well, except for the kurtosis. Nonetheless, we clearly see in the bottom panel of **Figure 5** that the distribution is right-skewed with more tail events on the left-hand side.

If we run the same model with the Gaussian assumption, by using a standard Brownian motion, *H* ¼ 0*:*5, we obtain a somewhat different result. The mean is of 0.94, the median of 0.85, the variance of 1.23, the skewness of 0.45 and the kurtosis of 2.64. This distribution is not right-skewed and has higher variance than the stylized facts. Hence, we conclude that a fractional Brownian motion with a Lévy process provide a better approach in order to model the inflation dynamics of the Eurozone.

Finally, we discuss the results of the simulation exercise with 1000 runs. In this simulation, we have specified our stochastic differential equation (Eq. (7)) as follows: *α* ¼ 1*:*7, *β* ¼ 0, *σ* ¼ 0*:*3, *μ* ¼ �2*:*0, *γ* ¼ 0*:*1, *λ* ¼ 0*:*00 and *H* ¼ 0*:*2. **Figure 6** represents in the top-panel the stochastic paths of all stochastic processes and in the bottom-panel the respective histogram. The numerical simulation yields a mean and median of approximately 1.7, a variance of 1.4 and negative skewness of �1.71.

runs of our simulation stable and strongly anchored at the 2.0% inflation target. Even in the worst negative or positive shock, inflation numbers do not reach levels

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

gent of the inflation target defined by the European Central Bank. Currently, inflation expectations are well anchored below the 2% level. Yet, our model simulation demonstrates that proposals to increase the inflation target, such as by Blanchard et al. [9], are highly risky because it leads to a de-anchoring of inflation. In the end, you might have higher volatility and the risk of de-anchored inflation expectations. The latter can create a strong upward bias in inflation rates out of the

That said, the stable and low inflation rates of the Eurozone are highly contin-

I thank the RRI for supporting my research. Moreover, I thank all conference participants for valuable comments and feedback to a preliminary version of this inbook article. Moreover, I thank the two anonymous referees for excellent comments and suggestions. All this improved the in-book article. All remaining errors remain

There is no external funding and no conflict of interest for this research article.

1 ESB Business School, Reutlingen University, Reutlingen, Germany

2 RRI Reutlingen Research Institute, Reutlingen, Germany

provided the original work is properly cited.

3 IFE Institute of Finance and Economics, Reutlingen, Germany

\*Address all correspondence to: bodo.herzog@reutlingen-university.de

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

persistently below 0 or above 4%.

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

control of a central bank.

**Acknowledgements**

my own responsibility.

**Conflict of interest**

**Author details**

Bodo Herzog1,2,3

**11**

#### **Figure 6.**

*Simulation of equation (7) with calibrated parameters. Top panel denotes all inflation processes and bottom panel the histogram. Source: B Herzog (2020).*

Last but not least, by running several simulations we find that inflation dynamics is with high likelihood in a range of [2, 5] in the Eurozone. Hence, even with severe positive or negative shocks the inflationary process is stable and anchored around the target level of 2%. Finally, in a scenario analysis, we set the mean-reverting level to the target rate of 4% as proposed by Blanchard et al. [9]. We find inflation dynamics is more volatile and still face deflationary levels during severe negative shocks. In that regard, a higher inflation target does not eliminate deflation events as with the target level of 2% today. Of course, the buffer towards deflation is greater if the inflation target is 4%. But economically, we proclaim that a higher inflation target creates a higher volatility and de-anchor inflation expectations subsequently. Consequently, increasing the inflation target is not free of any risk due to growing uncertainty about inflation expectations and price-stability in general.
