Abstract
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.
Highlights
Consider a system whose position or level is subject to random fluctuations and can be corrected by acting on its drift
We model this problem as a two-dimensional singular stochastic control problem
It is usually shown that the value function solves the associated dynamic programming principle in the classical sense and the free boundaries are characterized in terms of algebraic/integral equations
Summary
Consider a system whose position or level is subject to random fluctuations and can be corrected by acting on its drift. There is a large literature on two-dimensional degenerate boundedvariation stochastic control problems where the two components of the state-process are decoupled (see [2,16,20,23,24], and [35], among many others) In those works, it is usually shown that the value function solves the associated dynamic programming principle in the classical sense and the free boundaries are characterized in terms of algebraic/integral equations. It is usually shown that the value function solves the associated dynamic programming principle in the classical sense and the free boundaries are characterized in terms of algebraic/integral equations These results are obtained through various methods ranging from a “guess-and-verify” approach (see, e.g., [2,16], and [35]), viscosity theory ( [20]), the connection to optimal stopping ([31]), and the link to switching controls ([23] and [24]).
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