Kurushima Aiko

We consider a gradual-impulse control problem of continuous-time Markov decision processes, where the system performance is measured by the expectation of the exponential utility of the total cost. We show, under natural conditions on the system primitives, the existence of a deterministic stationary optimal policy out of a more general class of policies that allow multiple simultaneous impulses, randomized selection of impulses with random effects, and accumulation of jumps. After characterizing the value function using the optimality equation, we reduce the gradual-impulse control problem to an equivalent simple discrete-time Markov decision process, whose action space is the union of the sets of gradual and impulsive actions.

In this paper, we present a new monotone approximation of a given real-valued Carathéodory function on the product X × A of Borel spaces, where A is also compact. We demonstrate its application by providing a self-contained and elementary proof of a result of A. Nowak in discrete-time Markov decision processes.

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity lambda on some fixed time interval [0, T]. We assume a gamma prior density G(lambda)(r, 1/a) for lambda. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time s(i)((r)) onwards. The value of s(i)((r)) can be obtained for each r and i as the unique root of a deterministic equation.

This note studies a Poisson arrival selection problem for the full-information case with an unknown intensity lambda which has a Gamma prior density G(r, 1/a), where a > 0 and r is a natural number. For the no-information case with the same setting, the problem is monotone and the one-step look-ahead rule is an optimal stopping rule; in contrast, this note proves that the full-information case is not a monotone stopping problem.