  We study the Sharma-Parthasarathy stochastic two-body problem introduced by
N. Sharma and H. Parthasarathy in \cite{sharma}. In particular, we focus on the
preservation of some fundamental features of the classical two-body problem
like the Hamiltonian structure and first integrals in the stochastic case.
Numerical simulations are performed which illustrate the dynamical behaviour of
the osculating elements as the semi-major axis, the eccentricity and the
pericenter. We also derive a stochastic version of Gauss's equations in the
planar case.
