In this thesis, both the one dimensional and the multidimensional Hawkes-process are introduced. The first chapter contains the theoretical basis for this process. Both the exponential and the power law kernel are considered and important properties are presented. This includes e.g. the first and second order properties as well as the martingale representation and the relation between Hawkes processes and the Brownian motion. Furthermore, a second definition of the process via cluster representation is introduced. Subsequently, the maximum likelihood function of the process is derived. A special interest is taken in the appearance of the function for an exponential kernel. In the following chapter, the structure of the limit order book and the used data is explained. Said data is taken from the NASDAQ stock exchange. Statistical tests are introduced to check the data's eligibility concerning the simulation of Hawkes processes. The most important point is the transformation into a Poisson process. In the second part of the thesis, the theory is used to simulate a Hawkes process and compare the obtained results with the actual events. Therefor the parameters for an exponential kernel are estimated via maximum likelihood method and a simulation of the arrival times is done with the help of Ogata's thinning algorithm. The mentioned data is taken from the limit order book of the NASDAQ stock exchange. The analysis is done with different stocks which have a strong variation considering the number of arrival times. The obtained data are evaluated and checked for stability. In the end, the results are interpreted and possibilities for improvement are given.