Calculs stochastiques directs sur les trajectoires et propriétés de boreliens porteurs. (French) Zbl 0594.60060

Schwartz, Laurent

Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 271-326 (1984).

[For the entire collection see Zbl 0527.00020.]
Consider a general (vector) stochastic differential equation of the form \((*)\quad X=C+FX_-*Z,\) where the last term is the stochastic integral of \(FX_-\) relative to a semimartingale Z on a probability space (\(\Omega\),\(\Sigma\),\({\mathcal F}_ t,t\in {\mathbb{R}}^+,P)\). Here \({\mathcal F}_ t\uparrow \subset \Sigma\) is a filtration relative to which the Z-process is defined and F is a suitable matrix function. The usual method of solving this equation, with the initial condition \(X_ 0=C\) \((X_- (t)=X(t_-))\) is to employ the Picard (or the iterative) procedure with appropriate restrictions on F, to enable one to conclude that the approximants converge to the desired solution. In this form, Lipschitz conditions on F are the main hypotheses.
For such equations, K. Bichteler [Ann. Probab. 9, 49-89 (1981; Zbl 0458.60057)] showed that, with his a priori estimates on the growth of the solutions, the solution can be evaluated pathwise. Here the author has refined the methods further and considered conditions in order that the existence and essential uniqueness of solutions (i.e., the induced probability measures of any two solutions agree on their Borelian supports) when the conditions on F, Z are further relaxed, e.g., Z is not necessarily a cadlag process and F, depending on t, need not be predictable etc. To deal with these cases, the necessary theory is developed and then the work eventually leads to a study of the disintegration of probability measures into regular conditional measures extending some of the earlier seminal work of the author [J. Anal. Math. 26, 1-168 (1973; Zbl 0312.60025)].
The paper is a detailed exposition of the technical problems involved, and the results cannot be given in detail here. The following section headings will indicate the flavor of the work: 1. Intégrales stochastiques; 2. Equations différentielles stochastiques, cas de semi- martingales directrices discontinues; 3. Passage des semi-martingales de \(\Omega\) à \(W^ m\) [here \(W^ m\) is the space of the trajectories]; 4. Cas des semi-martingales directrices continues. Le flot; 5. Equations différentielles stochastiques localement lipschitziennes, avec temps de mort; 6. Le crochet [, ]. Equations différentielles stochastiques sur des variétés; 7. Désintégrations régulières.
The following from the introduction is relevant: ”Les démonstrations sont naturelles, parfois un peu longues, en général sans surprise; encore faut-il les faire, et on est heureux que cas résultats soient exacts. La meilleure preuve qu’on n’en est pas sûr à l’avance, est que certains résultats usuels qui donnent un \({\mathcal P}\) pour lequel ils restent vrais.” Here \({\mathcal P}\) is the family of probability measures governing the process, so that the conclusions are drawn for each member of \({\mathcal P}\). [In the notes on pages 317-318 read page number ”n” as \(''270+n.'']\)

Reviewer: M.M.Rao

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20	Stochastic integral equations
60G44	Martingales with continuous parameter

Keywords:

semimartingale; growth of the solutions; existence and essential uniqueness of solutions; Borelian supports

Citations:

Zbl 0527.00020; Zbl 0458.60057; Zbl 0312.60025