Stochastic Differential Equations
A diffusion can be thought of as a strong Markov process (in Rn) with continuous paths. Before the development of Itô’s theory of stochastic integration for Brownian motion, the primary method of studying diffusions was to study their transition semigroups; this was equivalent to studying the infinitesimal generators of their semigroups, which are partial differential operators. Thus Feller’s investigations of diffusions (for example) were actually investigations of partial differential equations, inspired by diffusions.
KeywordsBrownian Motion Stochastic Differential Equation Quadratic Variation Local Martingale Finite Variation
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- The extension of the (math) norm from martingales to semimartingales was implicit in Protter  and first formally proposed by Emery . A comprehensive account of this important norm for semimartingales can be found in Dellacherie-Meyer . Emery’s inequalities (Theorem 3) were first established in Emery , and later extended by Meyer .Google Scholar
- Existence and uniqueness of solutions of stochastic differential equations driven by general semimartingales was first established by Doléans-Dade  and Protter , building on the result for continuous semimartingales in Protter . Before this Kazamaki  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  in this regard. These results were improved and simplified by Doléans-Dade-Meyer  and Emery ; our approach is inspired by Emery . Métiver-Pellaumail  have an alternative approach. See also Métivier . Other treatments can be found in Doléans-Dade  and Jacod .Google Scholar
- The stability theory is due to Protter , Emery , and also to Métivier-Pellaumail . The semimartingale topology is due to Emery  and Métivier-Pellaumail . A pedagogic treatment is in Dellacherie-Meyer .Google Scholar
- The generalization of Fisk-Stratonovich integrals to semimartingales is due to Meyer . The treatment here of Fisk-Stratonovich differential equations is new. The idea of quadratic variation is due to Wiener . Theorem 18, which is a random Itô’s formula, appears in this form for the first time. It has an antecedent in Doss-Lenglart , and for a very general version (containing some quite interesting consequences), see Sznitman . Theorem 19 generalizes a result of Meyer , and Theorem 22 extends a result of Doss-Lenglart . Theorem 24 and its Corollary is from Ito . Theorem 25 is inspired by the work of Doss  (see also Ikeda-Watanabe  and Sussman ). The treatment of approximations of the Fisk-Stratonovich integrals was inspired by Yor . For an interesting application see Rootzen .Google Scholar
- The results of Sect. 6 are taken from Protter  and Çinlar-Jacod-Protter-Sharpe , A comprehensive pedagogic treatment when the Markov solutions are diffusions can be found in Stroock-Varadhan  or Williams  and Rogers-Williams .Google Scholar
- Work on flows of stochastic differential equations goes back to 1961 and the work of Blagovescenskii-Freidlin  who considered the Brownian case. For recent work on flows of stochastic differential equations, see Kunita [1–3], Ikeda-Watanabe  and the references therein. There are also flows results for the Brownian case in Gihman-Skorohod , but they are L2 rather than almost sure results. Much of our treatment is inspired by the work of Meyer  and that of Uppman [1,2] for the continuous case, however results are taken from other articles as well. For example, the example following Theorem 38 is due to Leandre , while the proof of Theorem 41, the non-confluence of solutions in the continuous case, is due to Emery ; an alternative proof is in Uppman . For the general (right continuous) case, we follow the work of Leandre . A similar result was obtained by Fujiwara-Kunita .Google Scholar