We consider dynamical systems that have emerged in financial studies and exhibit chaotic behaviour. The main purpose is to develop a data-based method for reconstruction of the trajectories of these systems. This methodology can then be used for prediction and control and it can also be utilized even if the dynamics of the system are unknown. To this end, we combine merits from Koopman operator theory, Extended Dynamic Mode Decomposition and Takens' embedding theorem. The result is a linear autoregressive model whose trajectories approximate the orbits of the original system. Finally, we enrich this method with machine learning techniques that can be used to train the autoregressive model.

We use the theory of Koopman (or composition) operators to study the dynamical system on $\mathbb{N}^*$ emerging from the well-known $3x+1$ (or Collatz) problem. The advantage of this approach is t...

The Athens Stock Exchange (ASE) is a dynamic financial market with complex interactions and inherent volatility. Traditional models often fall short in capturing the intricate dependencies and long memory effects observed in real-world financial data. In this study, we explore the application of fractional Brownian motion (fBm) to model stock price dynamics within the ASE, specifically utilizing the Athens General Composite (ATG) index. The ATG is considered a key barometer of the overall health of the Greek stock market. Investors and analysts monitor the index to gauge investor sentiment, economic trends, and potential investment opportunities in Greek companies. We find that the Hurst exponent falls outside the range typically associated with fractal Brownian motion. This, combined with the established non-normality of increments, disfavors both geometric Brownian motion and fractal Brownian motion models for the ATG index.

We apply methods from the Koopman operator theory, Extended Dynamic Mode Decomposition and machine learning in the study of business cycle models. We use a simple non-linear dynamical system whose main merit is that in the appropriate parameter space sector predicts intrinsically business cycles which in the phase space are structurally stable limit cycles. Our objective is to approximate this system with a finite dimensional linear model which is defined on some augmented state space. We approximate so the trajectories of the system and obtain an alternative non-perturbative description of the system which can be used for prediction and control. This approach can also be applied to other models as well as to real data.

We consider a data-driven method, which combines Koopman operator theory with Extended Dynamic Mode Decomposition. We apply this method to the hypergeometric equation which is the Fuchsian equation with three regular singular points. The space of solutions at any of its singular points is a two-dimensional linear vector space on the field of reals when the independent variable is restricted to take values in the real axis and the unknown function is restricted to be a real-valued function of a real variable. A basis of the linear vector space of solutions is spanned by the hypergeometric function and its products with appropriate powers of the independent variable or the logarithmic function depending on the roots of the indicial equation of the hypergeometric equation. With our work, we obtain a new representation of the fundamental solutions of the hypergeometric equation and relate them to the spectral analysis of the finite approximation of the Koopman operator associated with the hypergeometric equation. We expect that the usefulness of our results will come more to the fore when we extend our study into the complex domain.