**Dr. Mehmet Kenan Terzioğlu**

Associate Professor, Trakya University, Faculty of Economics and Administrative Sciences, Econometrics Department, Balkan Campus, Edirne, Turkey

#### **Gordana Djurovic**

University of Montenegro, Montenegro

**Chapter 1**

*Bodo Herzog*

**Abstract**

*b X*ð Þ , *<sup>t</sup> dBH*

lation output.

**1. Introduction**

not mean-reverting.

**1**

stochastic processes (**Figure 1**).

Lévy Processes

*<sup>t</sup>* , where *dB<sup>H</sup>*

Modeling Inflation Dynamics with

Fractional Brownian Motions and

The article studies a novel approach of inflation modeling in economics. We utilize a stochastic differential equation (SDE) of the form *dXt* ¼ *a X*ð Þ , *t dt* þ

ary dynamics. Standard economic models do not capture the stochastic nature of inflation in the Eurozone. Thus, we develop a new stochastic approach and take into consideration fractional Brownian motions as well as Lévy processes. The benefits of those stochastic processes are the modeling of interdependence and jumps, which is equally confirmed by empirical inflation data. The article defines and introduces the rules for stochastic and fractional processes and elucidates the stochastic simu-

**Keywords:** inflation, dynamics, modeling, stochastic differential equation,

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

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

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

fractional Brownian motion, Lévy process, jump-diffusion

*<sup>t</sup>* is a fractional Brownian motion in order to model inflation-
