Foundations of Modern Probability pp 515-536 | Cite as

# Semimartingales and General Stochastic Integration

Chapter

## Abstract

Predictable covariation and L^{2}-integral; semimartingale integral and covariation; general substitution rule; Doléans’ exponential and change of measure; norm and exponential inequalities; martingale integral; decomposition of semimartingales; quasi-martingales and stochastic integrators

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## References

- Doob (1953) conceived the idea of a stochastic integration theory for general
*L*^{2}-martingales, based on a suitable decomposition of continuous-time submartingales. Meyer’s (1962) proof of such a result opened the door to the*L*^{2}-theory, which was then developed by Courrège (1962–63) and Kunita and Watanabe (1967). The latter paper contains in particular a version of the general substitution rule. The integration theory was later extended in a series of papers by Meyer (1967) and Doléans-Dade and Meyer (1970) and reached its final form with the notes of Meyer (1976) and the books by Jacod (1979), Metivier and Pellaumail (1979), and Dellacherie and Meyer (1975–87).Google Scholar - The basic role of predictable processes as integrands was recognized by Meyer (1967). By contrast, semimartingales were originally introduced in an ad hoc manner by Doléans-Dade and Meyer (1970), and their basic preservation laws were only gradually recognized. In particular, Jacod (1975) used the general Girsanov theorem of Van Schuppen and Wong (1974) to show that the semimartingale property is preserved under absolutely continuous changes of the probability measure. The characterization of general stochastic integrators as semimartingales was obtained independently by Bichteler (1979) and Dellacherie (1980), in both cases with support from analysts.Google Scholar
- Quasimartingales were originally introduced by Fisk (1965) and Orey (1966). The decomposition of Rao (1969b) extends a result by Krickeberg (1956) for
*L*^{1}-bounded martingales. Yoeurp (1976) combined a notion of “stable subspaces” due to Kunita and Watanabe (1967) with the Hilbert space structure of*M*^{2}to obtain an orthogonal decomposition of*L*^{2}-martingales, equivalent to the decompositions in Theorem 26.14 and Proposition 26.16. Elaborating on those ideas, Meyer (1976) showed that the purely discontinuous component admits a representation as a sum of compensated jumps.Google Scholar - SDEs driven by general Lévy processes were already considered by Itô (1951b). The study of SDEs driven by general semimartingales was initiated by Doleans-Dade (1970), who obtained her exponential process as a solution to the equation in Theorem 26.8. The scope of the theory was later expanded by many authors, and a comprehensive account is given by Protter (1990).Google Scholar
- The martingale inequalities in Theorems 26.12 and 26.17 have ancient origins. Thus, a version of the latter result for independent random variables was proved by Kolmogorov (1929) and, in a sharper form, by Prohorov (1959). Their result was extended to discrete-time martingales by Johnson et al. (1985) and Hitczenko (1990). The present statements appeared in Kallenberg and Sztencel (1991).Google Scholar
- Early versions of the inequalities in Theorem 26.12 were proved by Khinchin (1923, 1924) for symmetric random walks and by Paley (1932) for Walsh series. A version for independent random variables was obtained by Marcinkiewicz and Zygmund (1937, 1938). The extension to discrete-time martingales is due to Burkholder (1966) for
*p*> 1 and to Davis (1970) for*p*= 1. The result was extended to continuous time by Burkholder et al. (1972), who also noted how the general result can be deduced from the statement for*p =*1. The present proof is a continuous-time version of Davis’ original argument.Google Scholar - Excellent introductions to semimartingales and stochastic integration are given by Dellacherie and Meyer (1975–87) and Jacod and Shiryaev (1987). Protter (1990) offers an interesting alternative approach, originally suggested by Meyer and by Dellacherie (1980). The book by Jacod (1979) remains a rich source of further information on the subject.Google Scholar

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