In this thesis, as mentioned above, we will deal with some approaches of fitting stochastic processes to the observed path of the mid price for a share. Theoretical models will be discussed as well as real stock data will be presented and analysed. For this purpose we use NASDAQ data from the web page lobsterdata.com. 20 big shares of the NASDAQ stock exchange will be analysed on the 16th November 2016, an arbitrarily chosen date by the author of this thesis. In the consecutive chapter theoretical approaches and results will be presented as well as developed mathematical limit order book models. The approaches used are basically very different, though. After that we will already discuss our used data, what is the structureof the data like and to which issues do we have to pay attention. Finally, the analysis and evaluation of the data is done. The following questions will be answered: How does the qualitative structure of the limit order book look like and does it vary over time? If yes, what may be the reasons for that? Does the structure of the LOB have an influence in the process of supply and demand and, subsequently, to the price formation process? These and other questions will be discussed in chapter 3. In the consecutive chapter 4 we will calibrate some stochastic processes, namely L ¿evy and Poisson processes. After this chapter a conclusion will be given where we will reflect on the things we have done and what the learnings are.