Abstract
We begin by describing a general Markov process running on continuous time and living in a topological space. The time parameter is the set of positive numbers, considered at first as just a linearly ordered set of indices. In the discrete case this is the set of positive integers and the corresponding discussion is given in Chapter 9 of the Course. Thus some of the proofs below are the same as for the discrete case. Only later when properties of sample functions are introduced will the continuity of time play an essential role. As for the living space we deal with a general one because topological properties of sets such as “open” and “compact” will be much used while specific Euclidean notions such as “interval” and “sphere” do not come into question until much later.
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Notes on Chapter 1
The basic notions of the Markov property, as well as optionality, in the discrete parameter case are treated in Chapters 8 and 9 of the Course. A number of proofs carry over to the continuous parameter case, without change.
Among the examples of Markov processes given here, only the case of Brownian motion will be developed in Chapter 4. But for dimension d = 1 the theory is somewhat special and will not be treated on its own merits. The case of Markov chains is historically the oldest, but its modern development is not covered by the general theory. It will only be mentioned here occasionally for peripheral illustrations. The class of spatially homogeneous Markov processes, sometimes referred to as Levy or additive processes, will be briefly described in §4.1.
Most of the material on optionality may be found in Chung and Doob [1] in a more general form. For deeper properties, which are sparingly used in this book, see Meyer [1]. The latter is somewhat dated but in certain respects more readable than the comprehensive new edition which is Dellacherie and Meyer [1].
It is no longer necessary to attribute the foundation of martingale theory to Doob, but his book [1] is obsolete especially for the treatment of the continuous parameter case. The review here borrows much from Meyer [1] and is confined to later needs, except for Theorems 6 and 7 which are given for the sake of illustration and general knowledge. Meyer’s proof of Theorem 5 initiated the method of projection in order to establish the optionality of the random time T defined in (33). This idea was later developed into a powerful methodology based on the two crfields mentioned at the end of the section. A very clear exposition of this “general theory” is given in Dellacherie [2]. A curious incident happened in the airplane from Zürich to Beijing in May of 1979. At the prodding of the author, Doob produced a new proof of Theorem 5 without using projection. Unfortunately it is not quite simple enough to be included here, so the interested reader must await its appearance in Doob’s forthcoming book.
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© 1982 Springer Science+Business Media New York
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Chung, K.L. (1982). Markov Process. In: Lectures from Markov Processes to Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol 249. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1776-1_1
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