1. The extension of the (math) norm from martingales to semimartingales was implicit in Protter [1] and first formally proposed by Emery [1]. A comprehensive account of this important norm for semimartingales can be found in Dellacherie-Meyer [2]. Emery’s inequalities (Theorem 3) were first established in Emery [1], and later extended by Meyer [12].
2. Existence and uniqueness of solutions of stochastic differential equations driven by general semimartingales was first established by Doléans-Dade [4] and Protter [2], building on the result for continuous semimartingales in Protter [1]. Before this Kazamaki [1] published a preliminary result, and of course the literature on stochastic differential equations driven by Brownian motion and Lebesgue measure, as well as Poisson processes, was extensive. See, for example, the book of Gihman-Skorohod [1] in this regard. These results were improved and simplified by Doléans-Dade-Meyer [2] and Emery [3]; our approach is inspired by Emery [3]. Métiver-Pellaumail [2] have an alternative approach. See also Métivier [1]. Other treatments can be found in Doléans-Dade [5] and Jacod [1].
3. The stability theory is due to Protter [4], Emery [3], and also to Métivier-Pellaumail [3]. The semimartingale topology is due to Emery [2] and Métivier-Pellaumail [3]. A pedagogic treatment is in Dellacherie-Meyer [2].
4. The generalization of Fisk-Stratonovich integrals to semimartingales is due to Meyer [8]. The treatment here of Fisk-Stratonovich differential equations is new. The idea of quadratic variation is due to Wiener [3]. Theorem 18, which is a random Itô’s formula, appears in this form for the first time. It has an antecedent in Doss-Lenglart [1], and for a very general version (containing some quite interesting consequences), see Sznitman [1]. Theorem 19 generalizes a result of Meyer [8], and Theorem 22 extends a result of Doss-Lenglart [1]. Theorem 24 and its Corollary is from Ito [7]. Theorem 25 is inspired by the work of Doss [1] (see also Ikeda-Watanabe [1] and Sussman [1]). The treatment of approximations of the Fisk-Stratonovich integrals was inspired by Yor [2]. For an interesting application see Rootzen [1].
5. The results of Sect. 6 are taken from Protter [3] and Çinlar-Jacod-Protter-Sharpe [1], A comprehensive pedagogic treatment when the Markov solutions are diffusions can be found in Stroock-Varadhan [1] or Williams [1] and Rogers-Williams [1].