Intermittent measurements frequently occur in practice, yet specific modeling is rarely used. Marked point processes (MPPs) provide a convenient framework to take into account such phenomenon. in this note, various discrete-time estimation problems are studied for a finite and homogeneous Markov chain observed by a marked point process. These problems, which could have significant applications in target tracking, manufacturing or communication theory, have never been studied in the literature. The quantities to be estimated are the state, the number of jumps and the occupation times. The identification of the chain transition matrix is also addressed via an expectation maximization (EM) procedure. Solutions, in the sense of the conditional distribution, are obtained by a change of probability measure and are shown to have convenient recursive forms. The efficiency of this new approach for sensor modeling is illustrated by the study of a linear markovian jump filtering problem where, in addition to a classical state observation, a mode MPP observation is assumed. A numerical example is given.