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Geometric approach to monotone stochastic control Chiarolla, Maria


The two main questions arising in singular control problems are the characterization of the boundary of the region of inaction A’ (i.e. the region where it is optimal to take no action) and the construction of an optimal control. Among the singular control problems the ones in which the class of admissible controls is restricted to the processes with monotone non-decreasing components, and the payoff functional does not depend explicitly on the control, are usually referred to as monotone follower, cheap control problems. We identify the free boundary ƌA1 of the two-dimensional monotone follower, cheap control problem under very mild conditions. We prove that if the region of inaction is of locally finite perimeter (LFP), then such a region can be replaced by a new region A1 having a more regular boundary. In fact, we show that the new free boundary is countably 1-rectifiable and it is also optimal to take no action in the larger set A1. Then we give conditions under which the hypothesis (LFP) holds; furthermore we obtain even higher regularity of the free boundary, namely C2α, except perhaps at a single corner point. This result is easily extended to the n-dimensional case. Under the additional hypothesis that the free boundary of the new region of inaction A1 satisfies a Lipschitz condition (LIP) in a small neighbourhood of the corner point, we construct a control k which acts only when the process is not in A1 and then only to move it instantaneously into A1. We show that k is the unique optimal control of the singular control problem in question. Finally we give conditions under which (LIP) is verified. All of these results hold in the n-dimensional case.

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