  This paper provides a practical approach to stochastic Lie systems, i.e.
stochastic differential equations whose general solutions can be written as a
function depending only on a generic family of particular solutions and some
constants, so as to emphasise their applications. We correct the known
stochastic Lie theorem characterising stochastic Lie systems, proving that,
contrary to previous claims, it satisfies the Malliavin's principle. Meanwhile,
we show that stochastic Lie systems admit new stochastic features in the Ito
approach. New generalisations of stochastic Lie systems, like the so-called
stochastic foliated Lie systems, are devised. Subsequently, we focus on
stochastic (foliated) Lie systems that can be studied as Hamiltonian systems
using different types of differential geometric structures. We study their
stability properties and we devise the basics of an energy-momentum method. A
stochastic Poisson coalgebra method is developed to derive superposition rules
for Hamiltonian stochastic Lie systems. Applications of our results are found
in coronavirus stochastic models, stochastic Lotka-Volterra systems, stochastic
SIS models of different types, etc. Our results improve previous approaches by
using stochastic differential equations instead of deterministic models
designed to grasp some of their stochastic features.
