TITLE: When is a continuous path stochastic process be representable as diffusion or Ito process?
QUESTION [9 upvotes]: When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?

REPLY [4 votes]: These types of questions are treated in great generality in the book
Rogers-Williams: Diffusions, Markov processes and martingales, Volume 1
One of the great results is due to Dynkin. Let $(X_t)_{t \ge 0}$ be a continuous Markov process whose semigroup is Feller-Dynkin (that is restricts to a strongly continuous semigroup on continuous functions vanishing at $\infty$). Then if the generator of $X$ contains the space of smooth and compactly supported functions, this generator is necessarily a second-order semi-elliptic differential operator and so $X$ is a diffusion process.