The need to obtain financial funds to pursue public utility investments implies the generation of Public-Private Partnership (PPP) projects. The PPP framework can involve risk-sharing mechanisms between public administration and private sector to encourage private investors to fund these projects. However, these risk-sharing mechanisms, such as Minimum Revenue Guarantee or Revenue Cap (RC), could generate opportunistic behaviors. For this reason, we can set this problem as a game in which government and private investors would act as players. This paper proposes a mathematical model to evaluate the PPP projects through a combination of Game Theory (GT) and Real Options Approach (ROA). The ROA is needed to price the uncertainty that affects PPP investments, and the GT captures the strategic interactions between public and private sectors. A case study on a wastewater treatment project is developed to apply the model we proposed.

This paper analyzes a parabolic operator $\mathcal{L}$ that generalizes several well-known operators commonly used in financial mathematics. We establish the existence and uniqueness of the Feller semigroup associated with $\mathcal{L}$ and derive its explicit analytical representation. The theoretical framework developed in this study provides a robust foundation for modeling stochastic processes relevant to financial markets. Furthermore, we apply these findings to energy market trading by developing specialized simulation algorithms and forecasting models. These methodologies were tested across all assets comprising the S&P 500 Energy Index, evaluating their predictive accuracy and effectiveness in capturing market dynamics. The empirical analysis demonstrated the practical advantages of employing generalized semigroups in modeling non-Gaussian market behaviors and extreme price fluctuations.