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The main goal of this paper is to establish existence, regularity anduniqueness results for the solution of a Hamilton-Jacobi-Bellman (HJB)equation, whose operator is an elliptic integro-differential operator. The HJBequation studied in this work arises in singular stochastic control problemswhere the state process is a controlled $d$-dimensional L\'evy process.

Solution to HJB equations with an elliptic integro-differential operator and gradient constraint