IV: 09, 77-107, LNM 124 (1970)
DOLÉANS-DADE, Catherine;
MEYER, Paul-André
Intégrales stochastiques par rapport aux martingales locales (
Martingale theory,
Stochastic calculus)
This is a continuation of Meyer
106, with a new complete exposition of the theory, and two substantial improvements: the filtration is general (while in
106 it was assumed free of fixed times of discontinuity) and the definition of semimartingales is the modern one (while in
106 they were the special semimartingales of nowadays). The change of variables formula is given in its full generality
Comment: The results of this paper have become classical, and are reproduced almost literally in Meyer
1017Keywords: Local martingales,
Stochastic integrals,
Change of variable formulaNature: Original
Retrieve article from Numdam
X: 17, 245-400, LNM 511 (1976)
MEYER, Paul-André
Un cours sur les intégrales stochastiques (6 chapters) (
Stochastic calculus,
Martingale theory,
General theory of processes)
This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$
Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also
1131. Now the material can be found in many books
Keywords: Increasing processes,
Stable subpaces,
Angle bracket,
Square bracket,
Stochastic integrals,
Optional stochastic integrals,
Previsible representation,
Change of variable formula,
Semimartingales,
Stochastic exponentials,
Multiplicative decomposition,
Fefferman inequality,
Davis inequality,
Stratonovich integrals,
Burkholder inequalities,
$BMO$,
Multiple stochastic integrals,
Girsanov's theoremNature: Exposition,
Original additions Retrieve article from Numdam
XV: 07, 118-141, LNM 850 (1981)
KUNITA, Hiroshi
Some extensions of Ito's formula (
Stochastic calculus)
The standard Ito formula expresses the composition of a smooth function $f$ with a continuous semimartingale as a stochastic integral, thus implying that the composition itself is a semimartingale. The extensions of Ito formula considered here deal with more complicated composition problems. The first one concerns a composition Let $(F(t, X_t)$ where $F(t,x)$ is a continuous semimartingale depending on a parameter $x\in
R^d$ and satisfying convenient regularity assumptions, and $X_t$ is a semimartingale. Typically $F(t,x)$ will be the flow of diffeomorphisms arising from a s.d.e. with the initial point $x$ as variable. Other examples concern the parallel transport of tensors along the paths of a flow of diffeomorphisms, or the pull-back of a tensor field by the flow itself. Such formulas (developed also by Bismut) are very useful tools of stochastic differential geometry
Keywords: Stochastic differential equations,
Flow of a s.d.e.,
Change of variable formula,
Stochastic parallel transportNature: Original Retrieve article from Numdam
XV: 09, 143-150, LNM 850 (1981)
FÖLLMER, Hans
Calcul d'Ito sans probabilités (
Stochastic calculus)
It is shown that if a deterministic continuous curve has a ``quadratic variation'' in a suitable sense (which however depends explicitly on a nested sequence of time subdivisions, for example the standard dyadic one), then it satisfies a deterministic ``Ito formula'' when composed with a twice differentiable function. Thus the only place where probability really appears in the derivation of Ito's formula is in the fact that, given any sequence of subdivisions, almost every path of a semimartingale admits a quadratic variation relative to this sequence (though no path may exist which has a quadratic variation relative to all sequences)
Comment: This subject is developed by T. Lyons' work on differential equations driven by non-smooth functions (in
Rev. Math. Iberoamericana 14, 1998)
Keywords: Stochastic integrals,
Change of variable formula,
Quadratic variationNature: Original Retrieve article from Numdam
XVI: 20, 234-237, LNM 920 (1982)
YOEURP, Chantha
Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (
Brownian motion,
Stochastic calculus)
A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$
Comment: See
1023,
1321Keywords: Multiplicative decomposition,
Change of variable formula,
Local timesNature: Original Retrieve article from Numdam
