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In this paper we study a representation problem first considered in a simplerversion by Bank and El Karoui [2004]. A key ingredient to this problem is arandom measure $\mu$ on the time axis which in the present paper is allowed tohave atoms. Such atoms turn out to not only pose serious technical challengesin the proof of the representation theorem, but actually have significantmeaning in its applications, for instance, in irreversible investment problems.These applications also suggest to study the problem for processes which aremeasurable with respect to a Meyer-$\sigma$-field that lies between thepredictable and the optional $\sigma$-field. Technically, our proof amounts toa delicate analysis of optimal stopping problems and the corresponding optimaldivided stopping times and we will show in a second application how an optimalstopping problem over divided stopping times can conversely be obtained fromthe solution of the representation problem.

On a Stochastic Representation Theorem for Meyer-measurable Processes and its Applications in Stochastic Optimal Control and Optimal Stopping