Some control problems with random intervention times

Abstract

We consider the problem of optimally tracking a Brownian motion by a sequence of impulse controls, in such a way to minimize the total expected cost that consists of a quadratic deviation cost and a proportional control cost. The main feature of our model is that the control can only be exerted at the arrival times of an exogenous uncontrolled Poisson process (signal). In other words, the set of possible intervention times are discrete, random and determined by the signal process (not by the decision maker). We discuss both the discounted problem and the ergodic problem, where explicit solutions can be found. We also derive the asymptotic behavior of the optimal control policies and the value functions as the intensity of the Poisson process goes to infinity, or roughly speaking, as the set of admissible controls goes from the discrete-time impulse control to the continuous-time bounded variation control.

Bounded Variation Control of Itô Diffusions with Exogenously Restricted Intervention Times

“…To show that the value F is attainable with the admissible policy ζ * , it suffices to show that J (x, ζ * ) ≥ F (x). First, since N jumps only upwards and F is nonnegative and nondecreasing, we find, using (20), that…”

“…The framework of [19] is elaborated further in [13]. For other related papers applying optimal control; see [20] and [18]. In [20], the author studies a classical optimal tracking problem for Brownian noise with quadratic running cost, under the assumption that the state variable can be controlled only at the jump times of N .…”

“…For other related papers applying optimal control; see [20] and [18]. In [20], the author studies a classical optimal tracking problem for Brownian noise with quadratic running cost, under the assumption that the state variable can be controlled only at the jump times of N . In [18], the authors study utility maximization when the stock price can be observed and traded only at the jump times of N , corresponding to the quotes on the market.…”

Hybrid Models and Switching Control with Constraints

“…Let us mention that reference related to optimal stopping with constraint include Lempa [20] and Liang [22], who studied particular cases of the model considered here, and that other classes of constraint have been considered, e.g., in Egloff and Leippold [12]. Moreover, for impulse control with constraint, relevant references include Brémaud [6,7], Liang and Wei [23], and Wang [37]. A different kind of constraint is considered in Costa et al [9], where the constraints are written as infinite horizon expected discounted costs.…”

On the Optimal Dividend Strategy in a Regime-Switching Diffusion Model

“…In [3] and our paper, the surplus processes are observed continuously, but we restrict ourselves to the case where the dividends can only be paid at some random discrete times. From this point of view, the problem considered in our paper is similar to [12].…”