Finite state Markov chains and prediction of stock market trends using real data

In this thesis we discuss finite state Markov chains, which are a special class of stochastic processes. They can be represented either by a graph or by a matrix [P]. The reader is first introduced to Markov chains and is then guided in their classification. Some relevant theorems are discussed. The results are used to explain when [P^n], the matrix obtained by taking the nth power of [P], converges as n approaches infinity. We start by studying the convergence in the case of [P] > 0 and we continue by focusing on two specific kinds of Markov chains: ergodic finite state chains and ergodic unichains. We then cover more general types of chains. In the end we give an example of how these tools can be used in the field of finance. We develop a model that predicts fluctuations in the prices of stocks and we apply it to the FTSE-MIB Index using data from Borsa Italiana.