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We derive the fluctuation-dissipation relation and explore its connectionwith the equipartition theorem and Maxwell-Boltzmann statistics through the useof different stochastic analytical techniques. Our first approach is the theoryof backward stochastic differential equations, which arises naturally in thiscontext, and facilitates the understanding of the interplay between theseclassical results of statistical mechanics. The second approach consists inderiving forward stochastic differential equations for the energy of anelectric system according to both It\^o and Stratonovich stochastic calculusrules. While the It\^o equation possesses a unique solution, which is thephysically relevant one, the Stratonovich equation admits this solution alongwith infinitely many more, none of which has a physical nature. Despite of thisfact, some, but not all of them, obey the fluctuation-dissipation relation.

Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus