  We derive the fluctuation-dissipation relation and explore its connection
with the equipartition theorem and Maxwell-Boltzmann statistics through the use
of different stochastic analytical techniques. Our first approach is the theory
of backward stochastic differential equations, which arises naturally in this
context, and facilitates the understanding of the interplay between these
classical results of statistical mechanics. The second approach consists in
deriving forward stochastic differential equations for the energy of an
electric system according to both It\^o and Stratonovich stochastic calculus
rules. While the It\^o equation possesses a unique solution, which is the
physically relevant one, the Stratonovich equation admits this solution along
with infinitely many more, none of which has a physical nature. Despite of this
fact, some, but not all of them, obey the fluctuation-dissipation relation.
