The development of integrated water resources management (IWRM) in the context of climate change and variability has created the need for an extension of mathematical analyses of hydrological system dynamics. The stationarity assumption of hydrological time series, which has been widely used in the past,  cannot be further advocated. The classical uncertainty modelling techniques based on probability theory cannot capture the multiple facets of present hydrological uncertainties. The objective of this study is to better capture the dynamics of the rainfall – runoff process. To this end, this research develops a rainfall runoff modelling approach that aims to capture the multiple sources and types of uncertainty in a single framework. The main assumption is that hydrological systems are non – linear dynamical systems which can be described by stochastic differential equations (SDE). The dynamics of the system is based on the Least Action Principle (LAP) as derived from Noether’s theorem. The inflow process is considered as a sum of deterministic and random components. The deterministic modelling of the river discharge in the Ouémé river basin (Benin, West Africa), using the hydrological model based on the least action principle (HyMoLAP), revealed that this model is suitable to simulate the daily dynamics of the river discharge. The stochastic formulation of HyMoLAP in terms of SDE allowed to better take into account the dynamics of the process and to explicitly show the proportion of the total variance of the discharge that is attributable to each source of uncertainties in the rainfall – runoff modelling. Then, the basic properties for the random component of rainfall are considered and the triple relationship between the structure of the inflowing rainfall, the corresponding SDE which describes the river basin and the associated Fokker – Planck equations (FPE) is analysed. The time – dependent probability distribution for the resulting discharge is obtained in the form of fundamental and approximate solutions of the FPE. A comparison is made between the time – dependent probability distributions and the empirical distribution of the outflow. The generalized FPE associated with the Langevin SDE describing the river basin is derived in terms of the transition probability distribution and characteristic function of the noise generating process. This equation provides a useful tool for studying the impact of various specific types of noises on the rainfall – runoff process.