This paper studies a class of non-Markovian singular stochastic controlproblems, for which we provide a novel probabilistic representation. Thesolution of such control problem is proved to identify with the solution of aZ-constrained BSDE, with dynamics associated to a non singular underlyingforward process. Due to the non-Markovian environment, our main argumentationrelies on the use of comparison arguments for path dependent PDEs. Ourrepresentation allows in particular to quantify the regularity of the solutionto the singular stochastic control problem in terms of the space and timeinitial data. Our framework also extends to the consideration of degeneratediffusions, leading to the representation of the solution as the infimum ofsolutions to Z-constrained BSDEs. As an application, we study the utilitymaximization problem with transaction costs for non-Markovian dynamics.
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