We develop a study of ignorability and conditions thereof for likelihood inference in the framework of stochastic processes. We define a coarsening model for processes which includes discrete-time observations as well as censored continuous-time observations and applies to continuous state-space processes as well as counting processes. For preparing the work we recall formulas for manipulating marginal and conditional likelihood ratios (which can apply to stochastic processes). Ignorability is defined in terms of local equality of two likelihood ratios. We give static conditions of ignorability and then dynamical conditions which are more interpretable. We illustrate the use of the dynamical conditions of ignorability in problems of censoring, missing data and joint modelling.