On singular stochastic control and optimal stopping of spectrally negative jump diffusions

We consider a broad class of singular stochastic control problems of spectrally negative jump diffusions in the presence of potentially nonlinear state-dependent exercise payoffs. We analyse these problems by relying on associated variational inequalities and state a set of sufficient conditions under which the value of the considered problems can be explicitly derived in terms of the increasing minimal r-harmonic map. We also present a set of inequalities bounding the value of the optimal policy and prove that increased policy flexibility increases both the value of the optimal strategy as well as the rate at which this value grows.