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In this paper we study a class of combined regular and singular stochasticcontrol problems that can be expressed as constrained BSDEs. In the Markoviancase, this reduces to a characterization through a PDE with gradientconstraint. But the BSDE formulation makes it possible to move beyond Markovianmodels and consider path-dependent problems. We also provide an approximationof the original control problem with standard BSDEs that yield acharacterization of approximately optimal values and controls.

BSDE formulation of combined regular and singular stochastic control problems