Applications of standard and Hamiltonian stochastic Lie systems

A stochastic Lie system on a manifold $M$ is a stochastic differential equation whose dynamics is described by a linear combination with functions depending on ${\mathbb{R}}^{\mathrm{\ell }}$-valued semi-martigales of vector fields on $M$ spanning a finite-dimensional Lie algebra. We analyse new examples of stochastic Lie systems and Hamiltonian stochastic Lie systems, and review and extend the coalgebra method for Hamiltonian stochastic Lie systems. We apply the theory to biological and epidemiological models, stochastic oscillators, stochastic Riccati equations, coronavirus models, stochastic Ermakov systems, etc.