A Controlled Jump Discounted Model with Constraints

The Transformation Method for Continuous-Time Markov Decision Processes

“…This random measure is predictable, and ν π (ω, {t} × S) = ν π (ω, [T ∞ , ∞[×S) = 0, see [14,16,17]. Therefore, by [14] or Chap.…”

“…In greater detail, analogously with [3,4,14,16,17], we present the construction of the stochastic basis (Ω, F , {F t } t≥0 , P π γ ) and the controlled process {ξ t , t ≥ 0} thereon of our interest. See also the discussion in [5] for an equivalent construction.…”

“…On the other hand, CTMDP optimization problems with the performance functional in the form of the right hand side of the above equality are prevailingly considered in the current literature about CTMDP optimization problems, see for instance, [4,13,17] and the references therein. In this connection, our CTMDP optimization problem (4) is in a more general form.…”

“…Now suppose on the opposite that there exists a policy π such that lim t→∞ for any non-negative functions f (x) and g(x) on R, the fourth inequality follows from (17), the sixth inequality is a result of Theorem 3.2 (when w 1 (x, a) = 0 and w 2 (x) = 1), and the last inequality is because of the condition of the statement. However, this yields the desired contradiction.…”

Absorbing Continuous-Time Markov Decision Processes with Total Cost Criteria

“…; see the examples in the monographs [13] and [25]. Two standard performance measures of a CTMDP are the (expected) long-run average costs [12], [14], [17], [24], [34], [39] and the (expected) total discounted costs [15], [27], [30]- [32]. The long-run average criteria are not appropriate for CTMDPs with transient behavior because in that case the long-run average costs will be zero for each policy.…”