XV: 30, 413-489, LNM 850 (1981)
SCHWARTZ, Laurent
Les semi-martingales formelles (
Stochastic calculus,
General theory of processes)
This is a natural development of the 1--1correspondence between semimartingales and $\sigma$-additive $L^0$-valued vector measures on the previsible $\sigma$-field, which satisfy a suitable boundedness property. What if boundedness is replaced by a $\sigma$-finiteness property? It turns out that these measures can be represented as formal stochastic integrals $H{\cdot} X$ where $X$ is a standard semimartingale, and $H$ is a (finitely valued, but possibly non-integrable) previsible process. The basic definition is quite elementary: $H{\cdot}X$ is an equivalence class of pairs $(H,X)$, where two pairs $(H,X)$ and $(K,Y)$ belong to the same class iff for some (hence for all) bounded previsible process $U>0$ such that $LH$ and $LK$ are bounded, the (usual) stochastic integrals $(UH){\cdot}X$ and $(UK){\cdot}Y$ are equal. (One may take for instance $U=1/(1{+}|H|{+}|K|)$.)\par As a consequence, the author gives an elegant and pedagogical characterization of the space $L(X)$ of all previsible processes integrable with respect to $X$ (introduced by Jacod,
1126; see also
1415,
1417 and
1424). This works just as well in the case when $X$ is vector-valued, and gives a new definition of vector stochastic integrals (see Galtchouk,
Proc. School-Seminar Vilnius, 1975, and Jacod
1419). \par Some topological considerations (that can be skipped if the reader is not interested in convergences of processes) are delicate to follow, specially since the theory of unbounded vector measures (in non-locally convex spaces!) requires much care and is difficult to locate in the literature
Keywords: Semimartingales,
Formal semimartingales,
Stochastic integralsNature: Original
Retrieve article from Numdam
XVI-S: 55, 1-150, LNM 921 (1982)
SCHWARTZ, Laurent
Géométrie différentielle du 2ème ordre, semi-martingales et équations différentielles stochastiques sur une variété différentielle Retrieve article from Numdam
XVIII: 24, 271-326, LNM 1059 (1984)
SCHWARTZ, Laurent
Calculs stochastiques directs sur les trajectoires et propriétés des boréliens porteurs Retrieve article from Numdam
XIX: 07, 91-112, LNM 1123 (1985)
SCHWARTZ, Laurent
Construction directe d'une diffusion sur une variété (
Stochastic differential geometry)
This seems to be the first use of Witney's embedding theorem to construct a process (a Brownian motion, a diffusion, a solution to some s.d.e.) in a manifold $M$ by embedding $M$ into some $
R^d$. Very general existence and uniqueness results are obtained
Comment: This method has since become standard in stochastic differential geometry; see for instance Émery's book
Stochastic Calculus in Manifolds (Springer, 1989)
Keywords: Diffusions in manifolds,
Stochastic differential equationsNature: Original Retrieve article from Numdam
XXIII: 27, 326-342, LNM 1372 (1989)
SCHWARTZ, Laurent
Le semi-groupe d'une diffusion en liaison avec les trajectoires Retrieve article from Numdam
XXIII: 28, 343-354, LNM 1372 (1989)
SCHWARTZ, Laurent
Convergence de la série de Picard pour les EDS Retrieve article from Numdam
XXIII: 29, 355-361, LNM 1372 (1989)
SCHWARTZ, Laurent
Quelques propriétés de la tribu accessible : les discontinuités d'un processus croissant intégrable et les discontinuités de sa projection prévisible duale Retrieve article from Numdam
XXIV: 38, 488-489, LNM 1426 (1990)
SCHWARTZ, Laurent
Quelques corrections et améliorations à mon article ``Le semi-groupe d'une diffusion en liaison avec les trajectoires'' Retrieve article from Numdam
XXVIII: 01, 1-20, LNM 1583 (1994)
SCHWARTZ, Laurent
Semi-martingales banachiques : le théorème des trois opérateurs Retrieve article from Numdam
XXX: 24, 369-370, LNM 1626 (1996)
SCHWARTZ, Laurent
Rectifications à ``Semi-martingales banachiques, le théorème des trois opérateurs'' (volume~XXVIII) Retrieve article from Numdam
