# Linear Programming Formulations of Singular Stochastic Control Problems

Richard Stockbridge^{*} and Thomas Kurtz

Many stochastic models include control actions which take effect instantly. For example, the decision maker may opt to reflect the process at a boundary so as to prevent it from becoming too large or too small. Actions may also induce instantaneous jumps in the process such as when fixed costs are charged on portfolio transactions resulting in decrease to the overall value of the portfolio. These type of actions are singular with respect to Lebesgue measure of time.\newline This talk will develop equivalent linear programming formulations of singular control problems having dynamics specified as the solutions to singularly controlled martingale problems; the dynamics also allow absolutely ocntinuous controls to be present in the model. A key to the equivalence is the definition of relaxed controls for the absolutely continuous controls, but more importantly for the singular controls as well. The reformulation depends on characterizing the expected stochastic behavior of the states and controls through appropriate expected occupation measures that satisfy adjoint equations corresponding to the cost criteria. These characterizations in terms of expected behavior are nevertheless rich enough to capture the full stochasticity of the processes through existence of relaxed solutions to the singular martingale problem. A consequence of these existence results is the equivalence of infinite-dimensional linear programming formulations for the stochastic problems. Under appropriate conditions, the existence of optimal measures and corresponding optimal controls in feedback form are obtained.

Mathematics Subject Classification: 93E20 60G44

Keywords: singular control; expected occupation measures; infinite-dimensional linear programming

Minisymposion: Stochastic Models, Control and Applications