Singular stochastic control in the presence of a state-dependent yield structure

We consider the determination of the optimal singular stochastic control for maximizing the expected cumulative revenue flows in the presence of a state-dependent marginal yield measuring the instantaneous returns accrued from irreversibly exerting the singular policy. As in standard models of singular stochastic control, the underlying stochastic process is assumed to evolve according to a regular linear diffusion. We derive the value of the optimal strategy by relying on a combination of stochastic calculus, the classical theory of diffusions, and non-linear programming. We state a set of usually satisfied conditions under which the optimal policy is to reflect the controlled process downwards at an optimal threshold satisfying an ordinary first-order necessary condition for an optimum. We also consider the comparative static properties of the value and state a set of sufficient conditions under which it is concave. As a consequence, we are able to state a set of sufficient conditions under which the sign of the relationship between the volatility of the process and the value is negative.

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