This talk is devoted to the long-term numerical integration of stochastic Hamiltonian and Poisson equations. The main goal is to study the conservation of Hamiltonians and Casimirs along the dynamics of suitable numerical approximations to the aforementioned systems, up to a large size of the time windows of integration. The methodology is the one the backward error analysis technique, that considers numerical flows as exact for derived modified equations associated to such stochastic differential problems. In particular, results on stochastic symplectic and Poisson integrators will be provided, establishing their capacity of preserving the invariant quantities of the aforementioned systems, up to a bounded error, for large time of integration. Finally, selected numerical examples and possible applications on the theoretical analysis on the direction of sustainable production processes will be also presented and discussed. The talk falls within the activities of the PRIN 2022 project 20229P2HEA “Stochastic numerical modelling for sustainable innovation”, CUP: E53D23017940001, granted by the Italian Ministry of University and Research within the framework of the Call relating to the scrolling of the final rankings of the PRIN 2022 call.