Exponential integrability of Itôʹs processes

In this note we prove the exponential integrability of super-norms of general Itôʹs processes under certain assumptions, and then apply it to the diffusion processes determined by stochastic differential equations. In particular, a conjecture in [Y. Hu, Exponential integrability of diffusion processes, in: Contemp. Math., vol. 234, 1999, pp. 75–84] is solved.