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II: 03, 34-42, LNM 51 (1968)

**DOLÉANS-DADE, Catherine**

Fonctionnelles additives parfaites (Markov processes)

The identity defining additive (or multiplicative) functionals involves an exceptional set depending on a continuous time $t$. If the exceptional set can be chosen independently of $t$, the functional is perfect. It is shown that every additive functional of a Hunt process admitting a reference measure has a perfect version

Comment: The existence of a reference measure was lifted by Dellacherie in 304. However, the whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

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II: 04, 43-74, LNM 51 (1968)

**DOLÉANS-DADE, Catherine**

Espaces $H^m$ sur les variétés, et applications aux équations aux dérivées partielles sur une variété compacte (Functional analysis)

An attempt to teach to the members of the seminar the basic facts of the analytic theory of diffusion processes

Keywords: Sobolev spaces, Second order elliptic equations

Nature: Exposition

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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 1017

Keywords: Local martingales, Stochastic integrals, Change of variable formula

Nature: Original

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IV: 19, 240-282, LNM 124 (1970)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**; **LETTA, Giorgio**; **MEYER, Paul-André**

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (*Comm. Pure Appl. Math.*, **22**, 1969), which constructs by a probability method a unique semigroup whose generator is an elliptic second order operator with continuous coefficients (the analytic approach either deals with operators in divergence form, or requires some Hölder condition). The contribution of G.~Letta nicely simplified the proof

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,*Multidimensional Diffusion Processes,* Springer 1979

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

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V: 12, 127-137, LNM 191 (1971)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**

Un contre-exemple au problème des laplaciens approchés (Martingale theory)

The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years

Comment: Problem originated in Meyer,*Ill. J. Math.*, **7**, 1963

Keywords: Submartingales, Supermartingales

Nature: Original

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V: 13, 138-140, LNM 191 (1971)

**DOLÉANS-DADE, Catherine**

Une martingale uniformément intégrable, non localement de carré intégrable (Martingale theory)

Now well known! This paper helped to set the basic notions of the theory

Keywords: Square integrable martingales

Nature: Original

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V: 14, 141-146, LNM 191 (1971)

**DOLÉANS-DADE, Catherine**

Intégrales stochastiques par rapport à une famille de probabilités (Stochastic calculus)

Given a family of probability laws on the same space, construct versions of stochastic integrals which do not depend on the law

Comment: Expanded by Stricker-Yor, Calcul stochastique dépendant d'un paramètre,*Z. für W-theorie,* **45**, 1978

Keywords: Stochastic integrals

Nature: Original

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XI: 24, 376-382, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Équations différentielles stochastiques (Stochastic calculus)

This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in*Zeit. für W-theorie,* **36**, 1976 and by Protter in *Ann. Prob.* **5**, 1977. The theory has become now so classical that the paper has only historical interest

Keywords: Stochastic differential equations, Semimartingales

Nature: Exposition, Original additions

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XI: 25, 383-389, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

Nature: Original

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XIII: 16, 204-215, LNM 721 (1979)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Un petit théorème de projection pour processus à deux indices (Several parameter processes)

This paper proves a previsible projection theorem in the case of two-parameter processes, with a two-parameter filtration satisfying the standard commutation property of Cairoli-Walsh. The idea is to apply successively the projection operation of 1315 to the two coordinates

Keywords: Previsible processes (several parameters), Previsible projections, Random measures

Nature: Original

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XIII: 28, 313-331, LNM 721 (1979)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Inégalités de normes avec poids (Martingale theory)

See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory

Comment: An exponent $1/\lambda$ is missing in formula (4), p.315

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

Retrieve article from Numdam

Fonctionnelles additives parfaites (Markov processes)

The identity defining additive (or multiplicative) functionals involves an exceptional set depending on a continuous time $t$. If the exceptional set can be chosen independently of $t$, the functional is perfect. It is shown that every additive functional of a Hunt process admitting a reference measure has a perfect version

Comment: The existence of a reference measure was lifted by Dellacherie in 304. However, the whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

Retrieve article from Numdam

II: 04, 43-74, LNM 51 (1968)

Espaces $H^m$ sur les variétés, et applications aux équations aux dérivées partielles sur une variété compacte (Functional analysis)

An attempt to teach to the members of the seminar the basic facts of the analytic theory of diffusion processes

Keywords: Sobolev spaces, Second order elliptic equations

Nature: Exposition

Retrieve article from Numdam

IV: 09, 77-107, LNM 124 (1970)

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 1017

Keywords: Local martingales, Stochastic integrals, Change of variable formula

Nature: Original

Retrieve article from Numdam

IV: 19, 240-282, LNM 124 (1970)

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

Retrieve article from Numdam

V: 12, 127-137, LNM 191 (1971)

Un contre-exemple au problème des laplaciens approchés (Martingale theory)

The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years

Comment: Problem originated in Meyer,

Keywords: Submartingales, Supermartingales

Nature: Original

Retrieve article from Numdam

V: 13, 138-140, LNM 191 (1971)

Une martingale uniformément intégrable, non localement de carré intégrable (Martingale theory)

Now well known! This paper helped to set the basic notions of the theory

Keywords: Square integrable martingales

Nature: Original

Retrieve article from Numdam

V: 14, 141-146, LNM 191 (1971)

Intégrales stochastiques par rapport à une famille de probabilités (Stochastic calculus)

Given a family of probability laws on the same space, construct versions of stochastic integrals which do not depend on the law

Comment: Expanded by Stricker-Yor, Calcul stochastique dépendant d'un paramètre,

Keywords: Stochastic integrals

Nature: Original

Retrieve article from Numdam

XI: 24, 376-382, LNM 581 (1977)

Équations différentielles stochastiques (Stochastic calculus)

This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in

Keywords: Stochastic differential equations, Semimartingales

Nature: Exposition, Original additions

Retrieve article from Numdam

XI: 25, 383-389, LNM 581 (1977)

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

Nature: Original

Retrieve article from Numdam

XIII: 16, 204-215, LNM 721 (1979)

Un petit théorème de projection pour processus à deux indices (Several parameter processes)

This paper proves a previsible projection theorem in the case of two-parameter processes, with a two-parameter filtration satisfying the standard commutation property of Cairoli-Walsh. The idea is to apply successively the projection operation of 1315 to the two coordinates

Keywords: Previsible processes (several parameters), Previsible projections, Random measures

Nature: Original

Retrieve article from Numdam

XIII: 28, 313-331, LNM 721 (1979)

Inégalités de normes avec poids (Martingale theory)

See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory

Comment: An exponent $1/\lambda$ is missing in formula (4), p.315

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

Retrieve article from Numdam