A multidimensional singular stochastic control problem on a finite time horizon

Marcin Boryc,Łukasz Kruk

doi:10.17951/a.2015.69.1.23

Abstract

A singular stochastic control problem in n dimensions with timedependent coefficients on a finite time horizon is considered. We show that the value function for this problem is a generalized solution of the corresponding HJB equation with locally bounded second derivatives with respect to the space variables and the first derivative with respect to time. Moreover, we prove that an optimal control exists and is unique.

Highlights

Singular stochastic control is a class of problems in which one is allowed to change the drift of a Markov process at a price proportional to the variation of the control used
Both C2-regularity of the value function and the characterization of the optimally controlled process have been extended to the case of singular control for the two-dimensional Brownian motion [14]
In this paper we consider a n-dimensional singular stochastic control problem on a finite time horizon in which state is governed by a linear stochastic differential equation with time-dependent coefficients, the running cost is convex and controls may act in any direction

Summary

MARCIN BORYC and ŁUKASZ KRUK

A multidimensional singular stochastic control problem on a finite time horizon Abstract. These estimates imply that the value function has locally bounded generalized derivatives of the second order with respect to the space variable and of the first order with respect to the time variable These properties are needed to consider the value function as a solution of the corresponding parabolic Hamilton–Jacobi–Bellman (HJB) equation in some generalized sense and to show existence and uniqueness of an optimal control. The analysis of [8] used the results from [11] as the starting point, so it is plausible that their analogs will be useful in proving the corresponding result on a finite time horizon Such a characterization would address a long-standing open problem on the structure of the optimal control in the case under consideration.

The solution of this problem satisfies s

We estimate A as follows n

Hence Aum conclude