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)


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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  1. 1.
    R. Boel, P. Varaiya, E. Wong:Martingales on Jump Processes Siam J. Control, Vol.13, No.51 (1975) I: Representation Results, 999–1021CrossRefGoogle Scholar
  2. 2.
    II: Applications, 1022–1061Google Scholar
  3. 3.
    M.H.A.Davis:The Representation of Martingales of Jump Processes Siam J.Control and Optimization, Vol.14, No.4, (july 1976)Google Scholar
  4. 4.
    M.H.A. Davis, P. Varaiya:Dynamic Programming Conditions For Partially Observable Stochastic Systems Siam J.Control 11, (1973), 226–261CrossRefGoogle Scholar
  5. 5.
    U.G. Hausmann:General Necessary Conditions for Optimal Control of Stochastic Systems University of British Columbia (1975)Google Scholar
  6. 6.
    M. Kohlmann:On Control of Jump Processes: A Martingale Approach, to be submittedGoogle Scholar
  7. 7.
    M. Kohlmann:A Martingale Approach to Optimal Control of Jump Processes, Lecture held at the II.Symp.on O.R., Aachen, (1977)Google Scholar
  8. 8.
    H.J. Kushner:Necessary conditions for continuous parameter stochastic optimization problems Siam J.Control, Vol.10, No.3, (1972), 550–565CrossRefGoogle Scholar
  9. 9.
    M. Loève:Probability Theory Van Nostrand Reinhold Company, N.Y.(1963)Google Scholar
  10. 10.
    P.A. Meyer:Probabilités et potentiel Hermann, Paris, (1966)Google Scholar
  11. 11.
    P.A. Meyer:Un cours sur les intégrales stochastiques Sém.Prob.Univ.Strasbourg(1974/75) Lecture Notes in Math., Springer VerlagGoogle Scholar
  12. 12.
    L.W. Neustadt:A General Theory of Extremals J.Computer and System Sciences 3, (1969), 57–92CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  1. 1.BonnGermany

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