Predictability, Compensation, and Excessive Functions
Accessible and predictable times; natural and predictable processes; Doob-Meyer decomposition; quasi-left-continuity; compensation of random measures; excessive and superharmonic functions; additive functionals as compensators; Riesz decomposition
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- The basic connection between superharmonic functions and supermartingales was established by Doob (1954), who also proved that compositions of excessive functions with Brownian motion are continuous. Doob further recognized the need for a general decomposition theorem for supermartingales, generalizing the elementary Lemma 7.10. Such a result was eventually proved by Meyer (1962, 1963), in the form of Lemma 25.7, after special decompositions in the Markovian context had been obtained by Volkonsky (1960) and Shur (1961). Meyer’s original proof was profound and clever. The present more elementary approach, based on Dunford’s (1939) weak compactness criterion, was devised by Rao (1969a). The extension to general submartingales was accomplished by Itô and Watanabe (1965) through the introduction of local martingales.Google Scholar
- Predictable and totally inaccessible times appear implicitly in the work of Blumenthal (1957) and Hunt (1957–58), in the context of quasi-left-continuity. A systematic study of optional times and their associated σ-fields was initiated by Chung and Doob (1965). The basic role of the predictable (7-field became clear after Doléans (1967a) had proved the equivalence between naturalness and predictability for increasing processes, thereby establishing the ultimate version of the Doob-Meyer decomposition. The moment inequality in Proposition 25.21 was obtained independently by Garsia (1973) and Neveu (1975) after a more special result had been proved by Burkholder et al. (1972). The theory of optional and predictable times and σ-fields was developed by Meyer (1966), Dellacherie (1972), and others into a “general theory of processes,” which has in many ways revolutionized modern probability.Google Scholar
- Natural compensators of optional times first appeared in reliability theory. More general compensators were later studied in the Markovian context by S. Watanabe (1964) under the name of “Levy systems.” Grigelionis (1971) and Jacod (1975) constructed the compensator of a general random measure and introduced the related “local characteristics” of a general semimartingale. Watanabe (1964) proved that a simple point process with a continuous and deterministic compensator is Poisson; a corresponding time-change result was obtained independently by Meyer (1971) and Papangelou (1972). The extension in Theorem 25.24 was given by Kallenberg (1990), and general versions of Proposition 25.27 appear in Rosinski and Woyczynski (1986) and Kallenberg (1992).Google Scholar
- An authoritative account of the general theory, including an elegant but less elementary projection approach to the Doob-Meyer decomposition due to Doléans, is given by Dellacherie and Meyer (1975–87). Useful introductions to the theory are contained in Elliott (1982) and Rogers and Williams (2000b). Our elementary proof of Lemma 25.10 uses ideas from Doob (1984). Blumenthal and Getoor (1968) remains a good general reference on additive functionals and their potentials. A detailed account of random measures and their compensators appears in Jacod and Shiryaev (1987). Applications to queuing theory are given by Brémaud (1981), Baccelli and Brémaud (2000), and Last and Brandt (1995).Google Scholar