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 utilize systems theory in the study of the implementation of non pharmaceutical strategies for the mitigation of the COVID-19 pandemic. We present two models. The first one is a model of predictive control with receding horizon and discontinuous actions of unknown costs for the implementation of adaptive triggering policies during the disease. This model is based on a periodic assessment of the peak of the pandemic (and, thus, of the health care demand) utilizing the latest data about the transmission and recovery rate of the disease. Consequently, the model seems to be suitable for discontinuous, non-mechanical (i.e. human) actions with unknown effectiveness, like those applied in the case of COVID-19. Secondly, we consider a feedback control problem in order to contain the pandemic at the capacity of the NHS (National Health System). As input parameter we consider the value $ p $ that reflects the intensity-effectiveness of the measures applied and as output the predicted maximum of infected people to be treated by NHS. The feedback control regulates $ p $ so that the number of infected people is manageable. Based on this approach, we address the following questions: (a) the limits of improvement of this approach; (b) the effectiveness of this approach; (c) the time horizon and timing of the application.

The paper addresses the problem of evaluating the Implicit McMillan degree$\delta{m}$ of $W^{-1}(s)$, where $W^{-1}(s)$ denotes the transfer function of a passive $RLC$ electrical network.1 The Implicit McMillan degree$\delta{m}$ specifies the minimum number of dynamic elements needed to completely characterize the passive $RLC$ network, i.e. an electrical network that contains only passive elements (capacitors, inductors and resistors) and associates it with the rank properties of the passive element matrices. A fact that in the circuit literature is intuitively accepted but not rigorously proved is that this degree must be equal to the minimum number of independent dynamical elements in the network Livada (2017, Implicit network descriptions of RLC networks and the problem of re engineering. Ph.D. Thesis, City, University of London) and Leventides et al. (2014, McMillan degree of impedance, admittance functions of RLC networks. In 21st International Symposium on Mathematical Theory of Networks and Systems. The Netherlands: Groningen). In this paper, we investigate this finding, showing that the maximum possible Implicit McMillan degree$\delta{m}$ of such networks is given by $rankL+rankC$ and that this value is reached when certain necessary and sufficient conditions are satisfied.1