We consider a standard Brownian motion whose drift can be increased or decreased in a possibly singular manner. The objective is to minimize an expected functional involving the time-integral of a running cost and the proportional costs of adjusting the drift. The resulting two-dimensional degenerate singular stochastic control problem has interconnected dynamics and it is solved by combining techniques of viscosity theory and free boundary problems. We provide a detailed description of the problem’s value function and of the geometry of the state space, which is split into three regions by two monotone curves. Our main result shows that those curves are continuously differentiable with locally Lipschitz derivative and solve a system of nonlinear ordinary differential equations.

我们考虑一个标准的布朗运动，其漂移可以以可能的奇异方式增加或减少。目的是使涉及运行成本和调整漂移的比例成本的时间积分的预期功能最小化。由此产生的二维简并奇异随机控制问题具有相互联系的动力学，并且可以通过将粘度理论和自由边界问题相结合来解决。我们提供了问题的值函数和状态空间几何的详细描述，状态空间由两条单调曲线分为三个区域。我们的主要结果表明，这些曲线可以用局部Lipschitz导数连续微分，并且可以求解非线性常微分方程组。