• Attractor
• Chaoticity
• Dynamical Systems
• Market
• Optimal Transport
• Population
• Random Measures
• Stability
• Volatility

The goal of the Master thesis is to generalize in more than one direction the population-market model proposed by Arlot, Marmi and Papini in [Arlot et al., 2019]. This model was itself an extension of the Birkeland-Yoccoz population model, studied in detail by Arlot in his Master thesis in the year 2004.

In Chapter 2, we briefly explain the theoretical results found in these two works: in both of them, the existence of a global attractor was proven. In [Arlot et al., 2019], the authors also proved the existence of at least a non-trivial periodic solution.

In Chapter 3, we introduce a stochastic component into the price equation. We firstly show how the previously found results are changed by this new setup. Due to the changes observed, in Section 3.2 we introduce the concepts of Random Attractor and of Random Invariant Measure. We conduct a detailed analysis in order to prove the existence of such asymptotic objects for the given dynamical system. A proof, made by H. Crauel, of a generalization of Prohorov Theorem to Random Measures, can be found in Appendix A. This is a key ingredient to show the existence of a Random Invariant Measure.

Finally, in Section 3.3 we show a result of convergence of the system to the deterministic one as the volatility tends to zero.

In Chapter 4, we instead try to simplify as much as possible the model through a discretization process. We first explain how the discretization is implemented and then we show the techniques we used to measure the similarities between the discretized system and the original one. One such instrument is the Optimal Transport Distance, which is explained, together with the associated algorithms, in Appendix B, where we report the contents of a work by M. Cuturi (2013).
We finally show in Section 4.3 some empirical observations that were made upon the randomized model.

A novel contribution of the thesis is the use of some general theorems, applied to the specific coupled Birkeland-Yoccoz model, to show the existence of a Random Attractor on the first component and of a Random Invariant Measure. Another contribution is the introduction of a specific parameter set, called H2, which is satisfyingly stable under reduction of dimensionality.