| Summary: | 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. === Science, Faculty of === Mathematics, Department of === Graduate
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