Speaker: Giorgio Ferrari, Universität Bielefeld
We present two continuous-time stochastic control problems involving controls of bounded-variation. Those problems are motivated by questions arising in macroeconomic theory - as the optimal control of inflation via interest rates - and in operations research - as the optimal management of an inventory having impact on the demand of a good – and their characteristic is that the drift of a diffusive component of the two-dimensional state variable is an affine function of a purely controlled process. The objective is to minimize an expected cost functional involving a running cost function and proportional costs of control.
By relying on a combination of techniques from viscosity theory and free-boundary analysis, weprovide the structure of the value function and we show that it satisfies a so-called second-order smooth-fit principle. Such a regularity is then exploited in order to determine a system of functional equations solved by the two monotone continuous curves (free boundaries) that split the control problem's state space in three connected regions. In one of the two considered problems, the free boundaries are even shown to be classical solutions to a system of first-order nonlinear ODEs.
This is based on joint works with Salvatore Federico (University of Siena) and Patrick Schuhmann (University of Bielefeld).