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Optimality Conditions in Optimal Control of Jump Processes — Extended Abstract

Conference paper
Part of the Proceedings in Operations Research 7 book series (ORP, volume 1977)

Abstract

The notations in this paper are mainly taken from [1 and 2] , as that paper is basic to ours.Let there be given the finite time interval [0, 1] -unless otherwise specified- and a familiy of probability spaces (Ω, Ft, P)t∈ [0, 1]with the usual properties. We are considering stochastic processes (xt, Ft, P) which are fundamental jump processes (f.j.p.) in the sense of [2] that are described by
$$ {P^x}(B,t): = \mathop \pi \limits_{s \leqslant t} [{x_{s - }} \ne {x_s}][{x_s} \in {\text{B}}] $$
where B is a subset of a measurable Blackwell space (Z,Z).Henceforth we assume Ft to be the completed ς-algebra generated by the f.j.p. (xt,P). Furthermore martingales (M1) ,twice integrable martingales (M2) ,local martingales (M loc 1 , M loc 2 ) ,integrable processes (A+) ,processes with integrable variation (A) and analogously (A loc + , Aloc) are denoted as in [2]. In order to define an integral with respect to elements from A we must define the associated sets of integrable functions. This will here only be done for a special case: Px(B,t) turns out to be a local semi martingale with some further properties.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  1. 1.BonnGermany

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