TV1982(1) - Vot,m XXVH THEORY OF PROBABILITY AND ITS APPLI...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
THEORY OF PROBABILITY Vot,,m XXVH A N D I T S A P P L I C A T I 0 N S m t 1982 NON-RANDOMIZED MARKOV AND SEMI-MARKOV STRATEGIES IN DYNAMIC PROGRAMMING E. A. FAINBERG (Translated by W. U. Sirk 1. Introduction In a non-homogeneous controllable Markov model with a total reward criterion, discrete time, infinite horizon and Borel spaces of states and controls, let a certain strategy 7r and an initial measure /x be given. In the paper the following two statements are proved: (a) (Theorem 3) for any K < +oo, there exists a non-randomized Markov strategy q such that > w(, 7r) if w(/x, rr)< +, 1) w (/., K if w(tx, 7r)= (b) (Theorem 4) for any measurable function K (x)< +oo given on a set of initial states X0, there exists a non-randomized semi-Markov strategy q’ such that, for any x X0, > J w(x, r) if w(x, 7r) < +o, (2) w(x, q ) [ K (x), if w (x, r) +c. The quantities w(/, r) and w(x, 7r) are the expectations of total reward in the case of the strategy 7r and initial measure/x, and initial state x, respectively. Controllable Markov models with Borel state spaces, as well as problems of existence of Markov and semi-Markov strategies in such models which majorize arbitrary strategies, were studied for the first time by Blackwall [1], [2]. These investigations were continued by Strauch [3], where three cases were considered: positive (P) and negative (N) dynamic programming, as well as dynamic programming with discounting (D). For the cases D and N it was proved, as one of the fundamental results of the investigation [3], Theorem 4.3], that non-randomized Markov strategies q and semi-Markov strategies q’ such that w (ix, q) -> w (/x, r) and w (x, o’) => w (x, r) for all initial states x exist. In all three cases, D, N and P, it was assumed in [3] that w (, r)< +o for all/x and zr, and in view of this the constant K and the function K (x) were not considered. For the case P (cf. [3], Theorem 4.4), existence of non-randomized Markov strategies q and semi-Markov strategies q’, such that w (, 0)-> w (/x, zr)-e and w(x, o’)>=w(x, zr)-e for all initial states x, was proved for any e >0. In [3] it 116
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
NON-RANDOMIZED MARKOV AND SEMI-MARKOV STRATEGIES 117 was pointed out that it is not known whether the last result is true for e 0. (We note that in the formulation of the problem it was assumed in [3] that the initial measure is concentrated at a single point. The case of an arbitrary initial measure/x, for the first time considered by Hinderer [4], does not introduce additional difficulties.) Homogeneous models were considered in [1]-[3]. The concept of non- homogeneous controllable models arose as a result of the investigations [5]-[7]. In [4], [8] and [9] a considerable part of the investigations [1]-[3] was extended to the case of non-homogeneous models, with a broader class of income functions being investigated in [4] and [9] than in [1]-[3]. For weak conditions the results on existence of a non-randomized Markov strategy in the non-homogeneous case, which majorizes an arbitrary strategy, is presented in [9] Chapt.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern