A mathematical model for the emergence of innovations

In the thesis, a recently proposed urn-based model with triggering, describing how novelties and innovations emerge in real systems, is studied. Through the urn process a sequence is generated. Relevant statistics can be studied such as how many distinct elements appear in the sequence until a certain time and the frequency with which each of the elements has been observed. These two statistics turn out to follow the Heaps' and Zipf's law, respectively and some heuristic arguments have been proposed in the literature to justify the emergence of these laws. One of the main contributions of this thesis consists in providing rigorous proofs for these results. This is achieved using stochastic approximation techniques, that allow one to approximate certain classes of stochastic processes through by ordinary differential equations. A second contribution consists in the extension of these results analyzing the number of elements that have appeared at least k times in the sequence, for arbitrary k. Such extensions allow one to consider design problems where the two main parameters of the model are chosen in such a way to optimise meaningful objectives. These analytical results are then supported and validated by numerical simulations, which also allow one to observe other possibly interesting behaviors and patterns. Finally, some modifications of the model are proposed in order to address specific issues that could better match certain phenomena observed in real systems.