Singular stochastic control and optimal stopping with partial information of jump diffusions
We study partial information, possibly non-Markovian, singular stochastic control of jump diffusions and obtain general maximum principles. The results are used to find connections between singular stochastic control, reflected BSDEs and optimal stopping in the partial information case. Mathematics Subject Classification 2010: 93E20, 60H07, 60H10, 60HXX, 60J75
The Relaxed Stochastic Maximum Principle in Singular Optimal Control of Diffusions
This paper studies optimal control of systems driven by stochastic differential equations, where the control variable has two components, the first being absolutely continuous and the second singular. Our main result is a stochastic maximum principle for relaxed controls, where the first part of the control is a measure valued process. To achieve this result, we establish first order optimality necessary conditions for strict controls by using strong perturbation on the absolutely continuous component of the control and a convex perturbation on the singular one. The proof of the main result is based on the strict maximum principle, Ekeland’s variational principle, and some stability properties of the trajectories and adjoint processes with respect to the control variable.
The stochastic maximum principle for a singular control problem
We consider a stochastic control problem in which the control has two components: the first being absolutely continuous, and the second singular. We assume linear dynamics, convex cost criterion and convex state constraint, and we allow the coefficients of the system to be random and the absolutely continuous component of the control to enter both the drift and diffusion coefficients. We do not impose any Lp -bounds on the control. We obtain for this model a Stochastic Maximum Principle in integral form. This is the first version of the Stochastic Maximum Principle that covers the Stochastic Singular Control Problem. When we assume, as in other versions of the Stochastic Maximum Principle, that the admissible controls are square-integrable, we obtain not only a necessary but also a sufficient condition for optimality. The mathematical tools are those of Stochastic Calculus and Convex Analysis.