  In this paper we study the exponential twist, i.e. a path-integral
exponential change of measure, of a Markovian reference probability measure
$\P$. This type of transformation naturally appears in variational
representation formulae originating from the theory of large deviations and can
be interpreted in some cases, as the solution of a specific stochastic control
problem. Under a very general Markovian assumption on $\P$, we fully
characterize the exponential twist probability measure as the solution of a
martingale problem and prove that it inherits the Markov property of the
reference measure. The ''generator'' of the martingale problem shows a drift
depending on a ''generalized gradient'' of some suitable ''value function''
$v$.
