Description
The present work is an extended version of a manuscript of a course which the author taught at the University of Hamburg during summer 1969. The main purpose has been to give a rigorous foundation of stochastic dynamic programming in a manner which makes the theory easily applicable to many different practical problems. We mention the following features which should serve our purpose. a) The theory is built up for non- stationery models, thus making it possible to treat e.g. dynamic programming under risk, dynamic programming under uncertainty, Markovian models, stationery models, and models with finite horizon from a unified point of view. b) We use that notion of optimality (p-optimality) which seems to be most appropriate for practical purposes. c) Since we restrict ourselves to the foundations, we did not include practical problems and ways to their numerical solution, but we give (cf.section 8) a number of problems which show the diversity of structures accessible to non stationery dynamic programming. The main sources were the papers of Blackwell (65), Strauch (66) and Maitra (68) on stationery models with general state and action spaces and the papers of Dynkin (65), Hinderer (67) and Sirjaev (67) on non- stationery models. A number of results should be new, whereas most theorems constitute extensions (usually from stationery models to non- stationery models) or analogues to known results. 1. Introduction and summary.- I. Countable state space.- 2. Decision models and definition of the problem.- 3. The principle of optimality and the optimality equation.- 4. Value iteration.- 5. Criteria of optimality and existence of $$bar{p} $$-optimal plans.- 6. Sufficient statistics, Markovian and stationery models.- 7. Models with incomplete information.- 8. Examples of special models.- 9. Randomized plans.- 10. Dynamic programming under uncertainty.- II. General state space.- 11. Decision models.- 12. Measure-theoretic and topological preparations.- 13. Universal measurability of the maximal conditional expected reward.- 14. The optimality equation.- 15. Substitution of randomized plans by deterministic plans.- 16. A generalization of the fixed point theorem for contractions.- 17. Criteria of optimality and existence of $$bar{p} $$-optimal plans.- 18. Sufficient statistics, Markovian and stationery models.- 19. Validity of the optimality equation without topological assumptions on state space and action space.- 20. Supplementary remarks.- A.Notions of optimality.- B.Some results for general sets of admissible plans.- C.A short summary of results of stochastic dynamic programming not treated in the present work.- Appendix 1. List of symbols and conventions.- 2. Some notions and auxiliary results from probability theory.- 3. Conditional distributions and expectations.- Literature.- Index of definitions.
