Abstract
We defined a semimartingale as a “good integrator” in Chap. II, and this led naturally to defining the stochastic integral as a limit of sums. To express an integral as a limit of sums requires some path smoothness of the integrands and we limited our attention to processes in L: the space of adapted processes with paths that are left continuous and have right limits. The space L is sufficient to prove Itô’s formula, the Girsanov-Meyer theorem, and it also suffices in some applications such as stochastic differential equations. But other uses, such as martingale representation theory or local times, require a larger space of integrands.
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Bibliographic Notes
The extension by continuity of stochastic integration from processes in L to predictable processes is presented here for essentially the first time. However the procedure was indicated earlier in Protter [6,7]. Other approaches which are closely related are given by Jacod [1] and Chou-Meyer-Stricker [1] (see an exposition in Dellacherie-Meyer [2, p. 381]).
The “𝓗” in the space 𝓗 2 of semimartingales comes from Hardy spaces, and this bears explaining. When the semimartingale X E 𝓗 2 is actually a martingale, the space of 𝓗p martingales has a theory analogous to that of Hardy spaces. While this is not the case for semimartingales, the 𝓗 p norms for semimartingales are, in a certain sense, a natural generalization of the 𝓗 p norms for martingales, and so the name has been preserved. The 𝓗 p norm was first introduced by Emery [1], though it was implicit in Protter
Most of the treatment of stochastic integration in Sect. 2 of this chapter is new, though essentially all of the results have been proved elsewhere with different methods. I have benefited greatly throughout by discussions with Svante Janson. Theorem 1 (that 𝓗 2 is a Banach space) is due to Emery [2]. The equivalence of the two pseudonorms for 𝓗 2 (the Corollary of Theorem 24) is originally due to Yor [7], while the fact that 𝓗 2 (Q) ⊂ 𝓗 2 (P) if $ is bounded is originally due to Lenglart [4]. Theorem 25 is originally due to Jacod [1], p. 228. The concept of a predictable σ-algebra and its importance to stochastic integration is due to Meyer [4]. The local behavior of the stochastic integral was first investigated by McShane for his integral [1], and then independently by Meyer [8], who established Theorem 26 and its Corollary for the semimartingale integral. The first dominated convergence theorem for the semimartingale integral appearing in print seems to be in Jacod [1].
The theory of martingale representation dates back to Itô’s work on multiple stochastic integrals [6], and the theory for M2 presented here is largely due to Kunita-Watanabe [1]. A more powerful theory (for M1) is presented in Dellacherie-Meyer [2].
The theory of stochastic integration depending on a parameter is due to the fundamental work of Stricker-Yor [1], who used a key idea of Doléans-Dade [1]. The Fubini theorems for stochastic integration have their origins in the book of Doob [1] and Kallianpur-Striebel [1] for the Ito integral, Kailath-Segall-Zakai [1] for martingale integrals, and Jacod [1] for semimartingales. The counterexample to a general Pubini theorem presented here is due to S. Janson.
The theory of semimartingale local time is of course abstracted from Brownian local time, which is due to Levy [1]. It was related to stochastic integration by Tanaka (see McKean [1]) for Brownian motion and the Ito integral. The theory presented here for semimartingale local time is due largely to Meyer [8]. See also Millar [1]. The measure theory needed for the rigorous development of semimartingale local time was developed by Stricker-Yor [1], and the Meyer-Itô formula (Theorem 51) was formally presented in Yor [6], as well as in Jacod [1]. Theorem 52 is due to Yor [4], and has been extended by Ouknine [1]. A more general version (but more complicated) is in Çinlar-Jacod-Protter-Sharpe [1]. The proof of Kolmogorov’s Lemma given here is from Meyer [14]. The results proved under Hypothesis A are all due to Yor [3] except the Bouleau-Yor formula [1]. See Bouleau [1] for more on this formula.
Azema’s martingale is of course due to Azema [1], though our presentation of it is new and it is due largely to Svante Janson. For many more interesting results concerning Azema’s martingale, see Azema-Yor [1], Emery [6], and Meyer [16].
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© 1990 Springer-Verlag Berlin Heidelberg
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Protter, P. (1990). General Stochastic Integration and Local Times. In: Stochastic Integration and Differential Equations. Applications of Mathematics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02619-9_5
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DOI: https://doi.org/10.1007/978-3-662-02619-9_5
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