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UBC Theses and Dissertations
The existence of optimal singular controls for stochastic differential equations Suo, Wulin
We study a singular control problem where the state process is governed by an Ito stochastic differential equation allowing both classical and singular coutrols. By reformulating the state equation as a martingale problem on an appropriate canonical space, it is shown, under mild continuity conditions on the data, that an optimal control exists. The dynamic programming principle for the problem is established through the method of conditioning and concatenation. Moreover, it is shown that there exists a family of optimal controls such that the corresponding states form a Markov process. When the data is Lipschitz continuous, the value function is shown to be uniformly con tinuous and to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman variational inequality. We also provide a description of the continuation region, the region in which the optimal state process is continuous, and we show that there exists a family of optimal controls which keeps the state inside the region after a possible initial jump. The last part is independent of the rest of the thesis. Through stretching of time, the singular control problem is transformed into a new problem that involves only classical control. Such problems are relatively well understood. As a result, it is shown that there exists an optimal control where the classical control variable is in Markovian form and the increment of the singular control variable on any time interval is adapted to the state process on the same time interval.
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