Abstract
In Chap. II we defined a semimartingale as a good integrator and we developed a theory of stochastic integration for integrands in L, the space of adapted processes with left continuous, right-limited paths. Such a space of integrands suffices to establish a change of variables formula (or “Itô’s formula”), and it also suffices for many applications, such as the study of stochastic differential equations. Nevertheless the space L is not general enough for the consideration of such important topics as local times and martingale representation theorems. We need a space of integrands analogous to measurable functions in the theory of Lebesgue integration; thus defining an integral as a limit of sums — which requires a degree of smoothness on the sample paths — is inadequate. In this chapter we lay the groundwork necessary for an extension of our space of integrands, and the stochastic integral is then extended in Chap. IV.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliographic Notes
The material of Chap. III comprises a large part of the core of the “general theory of processes” as is presented, for example, in Dellacherie [1]. We have, however, presented only the minimum required for our treatment of general stochastic integration and stochastic differential equations in Chaps. IV and V. We have also tried to keep the proofs as non-technical and as intuitive as possible. To this end we have presented K.M. Rao’s proof of the Doob-Meyer decomposition (Theorem 6), see Rao [1,2], rather than the usual Doléans-Dade measure approach. This involved the use of natural processes, although our definition is a new one which is equivalent to the original one of Meyer for increasing processes. In Sect. 8 we give an elementary proof that a bounded process of integrable variation is natural if and only if it is predictable, thus showing the equivalence of the approaches.
The Doob decomposition is from Doob [1], and the Doob-Meyer decomposition (Theorem 6) is due to Meyer [1,2]. Our proof is taken from K.M. Rao [1,2]. The theory of quasimartingales was developed by Fisk [1], Orey [1], K.M.Rao [2], Stricker [1], and Métivier-Pellaumail [1].
The Fundamental Theorem of Local Martingales is due to Jia-an Yan and appears in an article of Meyer [9]; it was also proved independently by Doléans-Dade [4].
The notion of special semimartingales and canonical decompositions is due to Meyer [8]. The Girsanov-Meyer theorem (Theorem 20) dates back to the 1954 work of Maruyama [1], though we don’t try to change its name here. The version presented here is due to Meyer [8] who extended the work of Girsanov [1]; see also Lenglart [1]. For more on how the stochastic calculus can be used in signal detection theory, see (for example) Wong [1].
The Bichteler-Dellacherie theorem (Theorem 22) is due independently to Bichteler [1,2] and Dellacherie [2]. It was proved in the late 1970’s but it didn’t appear in print until 1979 in the book of Jacod [1]. Many people have made contributions to this theorem, which had at least some of its origins in the work of Métivier-Pellaumail [3], Mokobodzki, Nikishin, Letta, and Lenglart. Our treatment was inspired by Meyer [13] and by Yan [1].
Our treatment of natural versus predictable processes is new, though the idea comes from Lenglart [2]. For a more customary treatment which includes the unbounded case, see (for example) Doob [2], pp. 483–487.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Protter, P. (1990). Semimartingales and Decomposable Processes. In: Stochastic Integration and Differential Equations. Applications of Mathematics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02619-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-662-02619-9_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-02621-2
Online ISBN: 978-3-662-02619-9
eBook Packages: Springer Book Archive