In this paper we discuss stochastic di erential delay equations with Markovian switching. Such an equation can be regarded as the result of several stochastic di erential delay equations switching from one to the others according to the movement of a Markov chain. One of the main aims of this paper is to investigate the exponential stability of the equations.
This paper deals with denumerable continuous-time Markov decision processes (MDP) with constraints. The optimality criterion to be minimized is expected discounted loss, while several constraints of the same type are imposed. The transition rates may be unbounded, the loss rates are allowed to be unbounded as well (from above and from below), and the policies may be history-dependent and randomized. Based on Kolmogorov's forward equation and Dynkin's formula, we remind the reader about the Bellman equation, introduce and study occupation measures, reformulate the optimization problem as a (primary) linear program, provide the form of optimal policies for a constrained optimization problem here, and establish the duality between the convex analytic approach and dynamic programming. Finally, a series of examples is given to illustrate all of our results.
This paper deals with discrete-time Markov decision processes (MDPs) under constraints where all the objectives have the same form of expected total cost over the infinite time horizon. The existence of an optimal control policy is discussed by using the convex analytic approach. We work under the assumptions that the state and action spaces are general Borel spaces, and that the model is nonnegative, semicontinuous, and there exists an admissible solution with finite cost for the associated linear program. It is worth noting that, in contrast to the classical results in the literature, our hypotheses do not require the MDP to be transient or absorbing. Our first result ensures the existence of an optimal solution to the linear program given by an occupation measure of the process generated by a randomized stationary policy. Moreover, it is shown that this randomized stationary policy provides an optimal solution to this Markov control problem. As a consequence, these results imply that the set of randomized stationary policies is a sufficient set for this optimal control problem. Finally, our last main result states that all optimal solutions of the linear program coincide on a special set with an optimal occupation measure generated by a randomized stationary policy. Several examples are presented to illustrate some theoretical issues and the possible applications of the results developed in the paper.
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