Quick search | Browse volumes | |

XVI: 01, 1-7, LNM 920 (1982)

**TALAGRAND, Michel**

Sur les résultats de Feyel concernant les épaisseurs

Retrieve article from Numdam

XVI: 02, 8-28, LNM 920 (1982)

**DELLACHERIE, Claude**; **FEYEL, Denis**; **MOKOBODZKI, Gabriel**

Intégrales de capacités fortement sous-additives

Retrieve article from Numdam

XVI: 03, 29-40, LNM 920 (1982)

**DELLACHERIE, Claude**

Appendice à l'exposé précédent

Retrieve article from Numdam

XVI: 04, 41-90, LNM 920 (1982)

**LONDON, R.R.**; **McKEAN, Henry P.**; **ROGERS, L.C.G.**; **WILLIAMS, David**

A martingale approach to some Wiener-Hopf problems (two parts)

Retrieve article from Numdam

XVI: 05, 91-94, LNM 920 (1982)

**WILLIAMS, David**

A potential-theoretic note on the quadratic Wiener-Hopf equation for Q-matrices

Retrieve article from Numdam

XVI: 06, 95-132, LNM 920 (1982)

**MEYER, Paul-André**

Note sur les processus d'Ornstein-Uhlenbeck (Malliavin's calculus)

With every Gaussian measure $\mu$ one can associate an Ornstein-Uhlenbeck semigroup, for which $\mu$ is a reversible invariant measure. When $\mu$ is Wiener's measure on ${\cal C}(**R**)$, this semigroup is a fundamental tool in Malliavin's own approach to the ``Malliavin calculus''. See for instance Stroock's exposition of it in *Math. Systems Theory,* **13**, 1981. With this semigroup one can associate its generator $L$ which plays the role of the classical Laplacian, and the positive bilinear functional $\Gamma(f,g)= L(fg)-fLg-gLf$---leaving aside domain problems for simplicity---sometimes called ``carré du champ'', which plays the role of the squared classical gradient. As in classical analysis, one can define it as $\sum_i \nabla_i f\nabla i g$, the derivatives being relative to an orthonormal basis of the Cameron-Martin space. We may define Sobolev-like spaces of order one in two ways: either by the fact that $Cf$ belongs to $L^p$, where $C=-\sqrt{-L}$ is the ``Cauchy generator'', or by the fact that $\sqrt{\Gamma(f,f)}$ belongs to $L^p$. A result which greatly simplifies the analytical part of the ``Malliavin calculus'' is the fact that both definitions are equivalent. This is the main topic of the paper, and its proof uses the Littlewood-Paley-Stein theory for semigroups as presented in 1010, 1510

Comment: An important problem is the extension to higher order Sobolev-like spaces. For instance, we could define the Sobolev space of order 2 either by the fact that $C^2f=-Lf$ belongs to $L^p$, and on the other hand define $\Gamma_2(f,g)=\sum_{ij} \nabla_i\nabla_j f \nabla_i\nabla_j g$ (derivatives of order 2) and ask that $\sqrt{\Gamma_2(f,f)}\in L^p$. For the equivalence of these two definitions and general higher order ones, see 1816, which anyhow contains many improvements over 1606. Also, proofs of these results have been given which do not involve Littlewood-Paley methods. For instance, Pisier has a proof which only uses the boundedness in $L^p$ of classical Riesz transforms.\par Another trend of research has been the correct definition of ``higher gradients'' within semigroup theory (the preceding definition of $\Gamma_2(f,g)$ makes use of the Gaussian structure). Bakry investigated the fundamental role of ``true'' $\Gamma_2$, the bilinear form $\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(Lf,g)$, which is positive in the case of the Ornstein-Uhlenbeck semigroup but is not always so. See 1909, 1910, 1912

Keywords: Ornstein-Uhlenbeck process, Gaussian measures, Littlewood-Paley theory, Hypercontractivity, Hermite polynomials, Riesz transforms, Test functions

Nature: Exposition, Original additions

Retrieve article from Numdam

XVI: 07, 133-133, LNM 920 (1982)

**MEYER, Paul-André**

Appendice : Un résultat de D. Williams (Malliavin's calculus)

This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as ``quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process

Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis

Nature: Exposition

Retrieve article from Numdam

XVI: 08, 134-137, LNM 920 (1982)

**BAKRY, Dominique**

Remarques sur le processus d'Ornstein-Uhlenbeck en dimension infinie (Malliavin's calculus, Several parameter processes)

A process taking values in a space of sample paths can be considered as a two parameter process. Considering in this way the Ornstein-Uhlenbeck process (1606) raises a few natural questions, like the commutation of conditional expectations relative to the two filtrations---which is shown to hold true

Keywords: Ornstein-Uhlenbeck process

Nature: Original

Retrieve article from Numdam

XVI: 09, 138-150, LNM 920 (1982)

**BAKRY, Dominique**; **MEYER, Paul-André**

Sur les inégalités de Sobolev logarithmiques (two parts) (Applications of martingale theory)

These two papers are variations on a paper of G.F. Feissner (*Trans. Amer Math. Soc.*, **210**, 1965). Let $\mu$ be a Gaussian measure, $P_t$ be the corresponding Ornstein-Uhlenbeck semigroup. Nelson's hypercontractivity theorem states (roughly) that $P_t$ is bounded from $L^p(\mu)$ to some $L^q(\mu)$ with $q\ge p$. In another celebrated paper, Gross showed this to be equivalent to a logarithmic Sobolev inequality, meaning that if a function $f$ is in $L^2$ as well as $Af$, where $A$ is the Ornstein-Uhlenbeck generator, then $f$ belongs to the Orlicz space $L^2Log_+L$. The starting point of Feissner was to translate this again as a result on the ``Riesz potentials'' of the semi-group (defined whenever $f\in L^2$ has integral $0$) $$R^{\alpha}={1\over \Gamma(\alpha)}\int_0^\infty t^{\alpha-1}P_t\,dt\;.$$ Note that $R^{\alpha}R^{\beta}=R^{\alpha+\beta}$. Then the theorem of Gross implies that $R^{1/2}$ is bounded from $L^2$ to $L^2Log_+L$. This suggests the following question: which are in general the smoothing properties of $R^\alpha$? (Feissner in fact considers a slightly different family of potentials).\par The complete result then is the following : for $\alpha$ complex, with real part $\ge0$, $R^\alpha$ is bounded from $L^pLog^r_+L$ to $L^pLog^{r+p\alpha}_+L$. The method uses complex interpolation between two cases: a generalization to Orlicz spaces of a result of Stein, when $\alpha$ is purely imaginary, and the case already known where $\alpha$ has real part $1/2$. The first of these two results, proved by martingale theory, is of a quite general nature

Keywords: Logarithmic Sobolev inequalities, Hypercontractivity, Gaussian measures, Riesz potentials

Nature: Original

Retrieve article from Numdam

XVI: 10, 151-152, LNM 920 (1982)

**MEYER, Paul-André**

Sur une inégalité de Stein (Applications of martingale theory)

In his book*Topics in harmonic analysis related to the Littlewood-Paley theory * (1970) Stein uses interpolation between two results, one of which is a discrete martingale inequality deduced from the Burkholder inequalities, whose precise statement we omit. This note states and proves directly the continuous time analogue of this inequality---a mere exercise in translation

Keywords: Littlewood-Paley theory, Martingale inequalities

Nature: Exposition, Original additions

Retrieve article from Numdam

XVI: 11, 153-158, LNM 920 (1982)

**MEYER, Paul-André**

Interpolation entre espaces d'Orlicz (Functional analysis)

This is an exposition of Calderon's complex interpolation method, in the case of moderate Orlicz spaces, aiming at its application in 1609

Keywords: Interpolation, Orlicz spaces, Moderate convex functions

Nature: Exposition

Retrieve article from Numdam

XVI: 12, 159-183, LNM 920 (1982)

**BRANCOVAN, Mihaï**; **BRONNER, François**; **PRIOURET, Pierre**

Grandes déviations pour certains systèmes différentiels aléatoires

Retrieve article from Numdam

XVI: 13, 184-200, LNM 920 (1982)

**PRIOURET, Pierre**

Remarques sur les petites perturbations de systèmes dynamiques

Retrieve article from Numdam

XVI: 14, 201-208, LNM 920 (1982)

**PERKINS, Edwin A.**

Local time and pathwise uniqueness for stochastic differential equations

Retrieve article from Numdam

XVI: 15, 209-211, LNM 920 (1982)

**BARLOW, Martin T.**

$L(B_t,t)$ is not a semimartingale (Brownian motion)

The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale

Keywords: Local times, Semimartingales

Nature: Original

Retrieve article from Numdam

XVI: 16, 212-212, LNM 920 (1982)

**WALSH, John B.**

A non-reversible semi-martingale (Stochastic calculus)

A simple example is given of a continuous semimartingale (a Brownian motion which stops at time $1$ and starts moving again at time $T>1$, $T$ encoding all the information up to time $1$) whose reversed process is not a semimartingale

Keywords: Semimartingales, Time reversal

Nature: Original

Retrieve article from Numdam

XVI: 17, 213-218, LNM 920 (1982)

**FALKNER, Neil**; **STRICKER, Christophe**; **YOR, Marc**

Temps d'arrêt riches et applications (General theory of processes)

This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see 1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper 1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$

Keywords: Stopping times, Local times, Semimartingales, Previsible processes

Nature: Original

Retrieve article from Numdam

XVI: 18, 219-220, LNM 920 (1982)

**STRICKER, Christophe**

Les intervalles de constance de $\langle X,X\rangle$ (Martingale theory, Stochastic calculus)

For a continuous (local) martingale $X$, the constancy intervals of $X$ and $<X,X>$ are exactly the same. What about general local martingales? It is proved that $X$ is constant on the constancy intervals of $<X,X>$, and the converse holds if $X$ has the previsible representation property

Keywords: Quadratic variation, Previsible representation

Nature: Original

Retrieve article from Numdam

XVI: 19, 221-233, LNM 920 (1982)

**YOR, Marc**

Application de la relation de domination à certains renforcements des inégalités de martingales (Martingale theory)

The domination relation (Lenglart 1977) between a positive, right-continuous process $X$ and a previsible increasing process $A$ holds whenever $E[X_T]\le E[A_T]$ at stopping times. It plays an important role in the paper 1404 of Lenglart-Lepingle-Pratelli on martingale inequalities. Here it is shown to imply a general inequality involving $X^\ast_{\infty}$ and $1/A_{\infty}$, from which follow a number of inequalities for a continuous local martingale $M$. Among them, estimates on the ratios of the three quantities $M^\ast_{\infty}$, $<M>_{\infty}$, $\sup_{a,t} L^a_t$. One can recover also the stronger version of Doob's inequality, proved by Pitman 1517

Comment: See an earlier paper of the author on this subject,*Stochastics,* **3**, 1979. The author mentions that part of the results were discovered slightly earlier by R.~Gundy

Keywords: Martingale inequalities, Domination inequalities

Nature: 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, 1321

Keywords: Multiplicative decomposition, Change of variable formula, Local times

Nature: Original

Retrieve article from Numdam

XVI: 21, 238-247, LNM 920 (1982)

**YOR, Marc**

Sur la transformée de Hilbert des temps locaux browniens et une extension de la formule d'Itô (Brownian motion)

This paper is about the application to the function $(x-a)\log|x-a|-(x-a)$ (whose second derivative is $1/x-a$) of the Ito-Tanaka formula; the last term then involves a formal Hilbert transform $\tilde L^a_t$ of the local time process $L^a_t$. Such processes had been defined by Ito and McKean, and studied by Yamada as examples of Fukushima's ``additive functionals of zero energy''. Here it is proved, as a consequence of a general theorem, that this process has a jointly continuous version---more precisely, Hölder continuous of all orders $<1/2$ in $a$ and in $t$

Comment: For a modern version with references see Yor,*Some Aspects of Brownian Motion II*, Birkhäuser 1997

Keywords: Local times, Hilbert transform, Ito formula

Nature: Original

Retrieve article from Numdam

XVI: 22, 248-256, LNM 920 (1982)

**JEULIN, Thierry**

Sur la convergence absolue de certaines intégrales (General theory of processes)

This paper is devoted to the a.s. absolute convergence of certain random integrals, a classical example of which is $\int_0^t ds/|B_s|^{\alpha}$ for Brownian motion starting from $0$. The author does not claim to prove deep results, but his technique of optional increasing reordering (réarrangement) of a process should be useful in other contexts too

Comment: This paper greatly simplifies a proof in the author's*Semimartingales et Grossissement de Filtrations,* LNM **833**, p.44

Keywords: Enlargement of filtrations

Nature: Original

Retrieve article from Numdam

XVI: 23, 257-267, LNM 920 (1982)

**FLIESS, Michel**; **NORMAND-CYROT, Dorothée**

Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T.~Chen (Stochastic calculus)

Consider a s.d.e. in a manifold, $dX_t=\sum_i A_i(X)\,dM^i_t$ (Stratonovich differentials), driven by continuous real semimartingales $M^i_t$, and where the $A_i$ have the geometrical nature of vector fields. Such an equation has a counterpart in which the $M^i(t)$ are arbitrary deterministic piecewise smooth functions, and if this equation can be solved by some deterministic machinery, then the s.d.e. can be solved too, just by making the input random. Thus a bridge is drawn between s.d.e.'s and problems of deterministic control theory. From this point of view, the complexity of the problem reflects that of the Lie algebra generated by the vector fields $A_i$. Assuming these fields are complete (i.e., generate true one-parameter groups on the manifold) and generate a finite dimensional Lie algebra (which then is the Lie algebra of a matrix group), the problem can be linearized. If the Lie algebra is nilpotent, the solution then can be expressed explicitly as a function of a finite number of iterated integrals of the driving processes (Chen integrals), and this provides the required ``deterministic machine''. It thus appears that results like those of Yamato (*Zeit. für W-Theorie,* **47**, 1979) do not really belong to probability theory

Comment: See Kunita's paper 1432

Keywords: Stochastic differential equations, Lie algebras, Chen's iterated integrals, Campbell-Hausdorff formula

Nature: Original

Retrieve article from Numdam

XVI: 24, 268-284, LNM 920 (1982)

**UPPMAN, Are**

Sur le flot d'une équation différentielle stochastique (Stochastic calculus)

This paper is a companion to 1506, devoted to the main results on the flow of a (Lipschitz) stochastic differential equation driven by continous semimartingales: non-confluence of solutions from different initial points, surjectivity of the mapping, smooth dependence on the initial conditions. The proofs have been greatly simplified

Keywords: Stochastic differential equations, Flow of a s.d.e., Injectivity

Nature: Exposition, Original additions

Retrieve article from Numdam

XVI: 25, 285-297, LNM 920 (1982)

**UPPMAN, Are**

Un théorème de Helly pour les surmartingales fortes (Martingale theory)

Provide the set of (optional) strong supermartingales $X$ of the class (D) with the topology of weak $L^1$--convergence of $X_T$ at each stopping time $T$. Then it is shown that any subset which belongs uniformly to the class (D) is relatively compact, also in the sequential sense of extracting convergent subsequences

Comment: This paper was suggested by a similar result of Mokobodzki for strongly supermedian functions in potential theory

Keywords: Supermartingales, Strong supermartingales

Nature: Original

Retrieve article from Numdam

XVI: 26, 298-313, LNM 920 (1982)

**DELLACHERIE, Claude**; **LENGLART, Érik**

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (General theory of processes)

This paper is a sequel to 1524. Let $\Theta$ be a*chronology,* i.e., a family of stopping times containing $0$ and $\infty$ and closed under the operations $\land,\lor$---examples are the family of all stopping times, and that of all deterministic stopping times. The general problem discussed is that of defining an optional process $X$ on $[0,\infty]$ such that for each $T\in\Theta$ $X_T$ is a.s. equal to a given r.v. (${\cal F}_T$-measurable). While in 1525 the discussion concerned supermartingales, it is extended here to processes which satisfy a semi-continuity condition from the right

Keywords: Stopping times

Nature: Original

Retrieve article from Numdam

XVI: 27, 314-318, LNM 920 (1982)

**LENGLART, Érik**

Sur le théorème de la convergence dominée (General theory of processes, Stochastic calculus)

Consider previsible processes $U^n,U$ such that $U^n_T\rightarrow U_T$ in some sense at bounded previsible times $T$. The problem discussed is whether stochastic integrals $\int U^n_s dX_s$ converge (in the same sense) to $\int U_sdX_s$. Under a domination hypothesis, the answer is shown to be positive if the convergence is either weak convergence in $L^1$, or convergence in probability. The existence of the limiting process $U$ is not assumed in the paper; it is proved by a modification of an argument of Mokobodzki for which see 1110

Keywords: Stopping times, Optional processes, Weak convergence, Stochastic integrals

Nature: Original

Retrieve article from Numdam

XVI: 28, 319-337, LNM 920 (1982)

**EAGLESON, G.K.**; **MÉMIN, Jean**

Sur la contiguïté de deux suites de mesures : généralisation d'un théorème de Kabanov-Liptser-Shiryayev

Retrieve article from Numdam

XVI: 29, 338-347, LNM 920 (1982)

**YAN, Jia-An**

À propos de l'intégrabilité uniforme des martingales exponentielles (Martingale theory)

Sufficient conditions are given for the uniform integrability of the exponential ${\cal E}(M)$, where $M$ is a local martingale with jumps $\ge-1$, refining older results of Lépingle and Mémin, and of the author. They involve the Lévy measure of the martingale

Comment: In the lemma p.339 delete the assumption $0<\beta$

Keywords: Exponential martingales

Nature: Original

Retrieve article from Numdam

XVI: 30, 348-354, LNM 920 (1982)

**HE, Sheng-Wu**; **WANG, Jia-Gang**

The total continuity of natural filtrations (General theory of processes)

Total continuity of a filtration ${\cal F}$ means that ${\cal F}_T={\cal F}_{T-}$ at every stopping time $T$, not necessarily previsible. It is shown that the filtration of a Lévy process without fixed discontinuities is totally continuous if and only if the jump size is a deterministic function of the jump time. Similarly, the natural filtration of a quasi-left continuous jump process is totally continuous if and only if the size of the $n$-th jump is a deterministic function of the jump times up to the $n$-th. It is shown that under the usual (here called ``strong'') previsible representation property, quasi-left continuity of the filtration implies total continuity

Keywords: Filtrations, Independent increments, Previsible representation, Total continuity, Lévy processes

Nature: Original

Retrieve article from Numdam

XVI: 31, 355-369, LNM 920 (1982)

**BAKRY, Dominique**

Semimartingales à deux indices

Retrieve article from Numdam

XVI: 32, 370-379, LNM 920 (1982)

**ZHENG, Wei-An**

Semimartingales in predictable random open sets

Retrieve article from Numdam

XVI: 33, 380-383, LNM 920 (1982)

**ABOULAÏCH, Rajae**

Intégrales stochastiques généralisées

Retrieve article from Numdam

XVI: 34, 384-391, LNM 920 (1982)

**KARANDIKAR, Rajeeva L.**

A.s. approximation results for multiplicative stochastic integrals

Retrieve article from Numdam

XVI: 35, 392-399, LNM 920 (1982)

**MEILIJSON, Isaac**

There exists no ultimate solution to Skorokhod's problem

Retrieve article from Numdam

XVI: 36, 400-408, LNM 920 (1982)

**EL KAROUI, Nicole**

Une propriété de domination de l'enveloppe de Snell des semimartingales fortes

Retrieve article from Numdam

XVI: 37, 409-411, LNM 920 (1982)

**CHOU, Ching Sung**

Une remarque sur l'approximation des solutions d'e.d.s.

Retrieve article from Numdam

XVI: 38, 412-441, LNM 920 (1982)

**KAWABATA, Shigetoku**; **YAMADA, Toshio**

On some limit theorems for solutions of stochastic differential equations

Retrieve article from Numdam

XVI: 39, 442-446, LNM 920 (1982)

**JACOD, Jean**

Équations différentielles linéaires : la méthode de variation des constantes

Retrieve article from Numdam

XVI: 40, 447-450, LNM 920 (1982)

**JACOD, Jean**; **PROTTER, Philip**

Quelques remarques sur un nouveau type d'équations différentielles stochastiques

Retrieve article from Numdam

XVI: 41, 451-468, LNM 920 (1982)

**PROTTER, Philip**

Stochastic differential equations with feedback in the differentials

Retrieve article from Numdam

XVI: 42, 469-489, LNM 920 (1982)

**PELLAUMAIL, Jean**

Règle maximale

Retrieve article from Numdam

XVI: 43, 490-502, LNM 920 (1982)

**MÉTIVIER, Michel**

Pathwise differentiability with respect to a parameter of solutions of stochastic differential equations

Retrieve article from Numdam

XVI: 44, 503-508, LNM 920 (1982)

**MEYER, Paul-André**

Résultats d'Atkinson sur les processus de Markov

Retrieve article from Numdam

XVI: 45, 509-514, LNM 920 (1982)

**GRAVERSEN, Svend Erik**; **RAO, Murali**

Hypothesis (B) of Hunt

Retrieve article from Numdam

XVI: 46, 515-518, LNM 920 (1982)

**GLOVER, Joseph**

An extension of Motoo's theorem

Retrieve article from Numdam

XVI: 47, 519-543, LNM 920 (1982)

**GHOUSSOUB, Nassif**

An integral representation of randomized probabilities and its applications

Retrieve article from Numdam

XVI: 48, 544-569, LNM 920 (1982)

**CHEVET, Simone**

Topologies métrisables rendant continues les trajectoires d'un processus

Retrieve article from Numdam

XVI: 49, 570-580, LNM 920 (1982)

**CHATTERJI, Shrishti Dhav**; **RAMASWAMY, S.**

Mesures gaussiennes et mesures produits

Retrieve article from Numdam

XVI: 50, 581-601, LNM 920 (1982)

**EHRHARD, Antoine**

Sur la densité du maximum d'une fonction aléatoire gaussienne

Retrieve article from Numdam

XVI: 51, 602-608, LNM 920 (1982)

**HEINKEL, Bernard**

Sur la loi du logarithme itéré dans les espaces réflexifs

Retrieve article from Numdam

XVI: 52, 609-622, LNM 920 (1982)

**LEDOUX, Michel**

La loi du logarithme itéré pour les variables aléatoires prégaussiennes à valeurs dans un espace de Banach à norme régulière

Retrieve article from Numdam

XVI: 53, 623-623, LNM 920 (1982)

**MEYER, Paul-André**

Correction au Séminaire XV

Retrieve article from Numdam

XVI: 54, 623-623, LNM 920 (1982)

**MEYER, Paul-André**

Addendum au Séminaire XV

Retrieve article from Numdam

Sur les résultats de Feyel concernant les épaisseurs

Retrieve article from Numdam

XVI: 02, 8-28, LNM 920 (1982)

Intégrales de capacités fortement sous-additives

Retrieve article from Numdam

XVI: 03, 29-40, LNM 920 (1982)

Appendice à l'exposé précédent

Retrieve article from Numdam

XVI: 04, 41-90, LNM 920 (1982)

A martingale approach to some Wiener-Hopf problems (two parts)

Retrieve article from Numdam

XVI: 05, 91-94, LNM 920 (1982)

A potential-theoretic note on the quadratic Wiener-Hopf equation for Q-matrices

Retrieve article from Numdam

XVI: 06, 95-132, LNM 920 (1982)

Note sur les processus d'Ornstein-Uhlenbeck (Malliavin's calculus)

With every Gaussian measure $\mu$ one can associate an Ornstein-Uhlenbeck semigroup, for which $\mu$ is a reversible invariant measure. When $\mu$ is Wiener's measure on ${\cal C}(

Comment: An important problem is the extension to higher order Sobolev-like spaces. For instance, we could define the Sobolev space of order 2 either by the fact that $C^2f=-Lf$ belongs to $L^p$, and on the other hand define $\Gamma_2(f,g)=\sum_{ij} \nabla_i\nabla_j f \nabla_i\nabla_j g$ (derivatives of order 2) and ask that $\sqrt{\Gamma_2(f,f)}\in L^p$. For the equivalence of these two definitions and general higher order ones, see 1816, which anyhow contains many improvements over 1606. Also, proofs of these results have been given which do not involve Littlewood-Paley methods. For instance, Pisier has a proof which only uses the boundedness in $L^p$ of classical Riesz transforms.\par Another trend of research has been the correct definition of ``higher gradients'' within semigroup theory (the preceding definition of $\Gamma_2(f,g)$ makes use of the Gaussian structure). Bakry investigated the fundamental role of ``true'' $\Gamma_2$, the bilinear form $\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(Lf,g)$, which is positive in the case of the Ornstein-Uhlenbeck semigroup but is not always so. See 1909, 1910, 1912

Keywords: Ornstein-Uhlenbeck process, Gaussian measures, Littlewood-Paley theory, Hypercontractivity, Hermite polynomials, Riesz transforms, Test functions

Nature: Exposition, Original additions

Retrieve article from Numdam

XVI: 07, 133-133, LNM 920 (1982)

Appendice : Un résultat de D. Williams (Malliavin's calculus)

This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as ``quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process

Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis

Nature: Exposition

Retrieve article from Numdam

XVI: 08, 134-137, LNM 920 (1982)

Remarques sur le processus d'Ornstein-Uhlenbeck en dimension infinie (Malliavin's calculus, Several parameter processes)

A process taking values in a space of sample paths can be considered as a two parameter process. Considering in this way the Ornstein-Uhlenbeck process (1606) raises a few natural questions, like the commutation of conditional expectations relative to the two filtrations---which is shown to hold true

Keywords: Ornstein-Uhlenbeck process

Nature: Original

Retrieve article from Numdam

XVI: 09, 138-150, LNM 920 (1982)

Sur les inégalités de Sobolev logarithmiques (two parts) (Applications of martingale theory)

These two papers are variations on a paper of G.F. Feissner (

Keywords: Logarithmic Sobolev inequalities, Hypercontractivity, Gaussian measures, Riesz potentials

Nature: Original

Retrieve article from Numdam

XVI: 10, 151-152, LNM 920 (1982)

Sur une inégalité de Stein (Applications of martingale theory)

In his book

Keywords: Littlewood-Paley theory, Martingale inequalities

Nature: Exposition, Original additions

Retrieve article from Numdam

XVI: 11, 153-158, LNM 920 (1982)

Interpolation entre espaces d'Orlicz (Functional analysis)

This is an exposition of Calderon's complex interpolation method, in the case of moderate Orlicz spaces, aiming at its application in 1609

Keywords: Interpolation, Orlicz spaces, Moderate convex functions

Nature: Exposition

Retrieve article from Numdam

XVI: 12, 159-183, LNM 920 (1982)

Grandes déviations pour certains systèmes différentiels aléatoires

Retrieve article from Numdam

XVI: 13, 184-200, LNM 920 (1982)

Remarques sur les petites perturbations de systèmes dynamiques

Retrieve article from Numdam

XVI: 14, 201-208, LNM 920 (1982)

Local time and pathwise uniqueness for stochastic differential equations

Retrieve article from Numdam

XVI: 15, 209-211, LNM 920 (1982)

$L(B_t,t)$ is not a semimartingale (Brownian motion)

The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale

Keywords: Local times, Semimartingales

Nature: Original

Retrieve article from Numdam

XVI: 16, 212-212, LNM 920 (1982)

A non-reversible semi-martingale (Stochastic calculus)

A simple example is given of a continuous semimartingale (a Brownian motion which stops at time $1$ and starts moving again at time $T>1$, $T$ encoding all the information up to time $1$) whose reversed process is not a semimartingale

Keywords: Semimartingales, Time reversal

Nature: Original

Retrieve article from Numdam

XVI: 17, 213-218, LNM 920 (1982)

Temps d'arrêt riches et applications (General theory of processes)

This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see 1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper 1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$

Keywords: Stopping times, Local times, Semimartingales, Previsible processes

Nature: Original

Retrieve article from Numdam

XVI: 18, 219-220, LNM 920 (1982)

Les intervalles de constance de $\langle X,X\rangle$ (Martingale theory, Stochastic calculus)

For a continuous (local) martingale $X$, the constancy intervals of $X$ and $<X,X>$ are exactly the same. What about general local martingales? It is proved that $X$ is constant on the constancy intervals of $<X,X>$, and the converse holds if $X$ has the previsible representation property

Keywords: Quadratic variation, Previsible representation

Nature: Original

Retrieve article from Numdam

XVI: 19, 221-233, LNM 920 (1982)

Application de la relation de domination à certains renforcements des inégalités de martingales (Martingale theory)

The domination relation (Lenglart 1977) between a positive, right-continuous process $X$ and a previsible increasing process $A$ holds whenever $E[X_T]\le E[A_T]$ at stopping times. It plays an important role in the paper 1404 of Lenglart-Lepingle-Pratelli on martingale inequalities. Here it is shown to imply a general inequality involving $X^\ast_{\infty}$ and $1/A_{\infty}$, from which follow a number of inequalities for a continuous local martingale $M$. Among them, estimates on the ratios of the three quantities $M^\ast_{\infty}$, $<M>_{\infty}$, $\sup_{a,t} L^a_t$. One can recover also the stronger version of Doob's inequality, proved by Pitman 1517

Comment: See an earlier paper of the author on this subject,

Keywords: Martingale inequalities, Domination inequalities

Nature: Original

Retrieve article from Numdam

XVI: 20, 234-237, LNM 920 (1982)

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, 1321

Keywords: Multiplicative decomposition, Change of variable formula, Local times

Nature: Original

Retrieve article from Numdam

XVI: 21, 238-247, LNM 920 (1982)

Sur la transformée de Hilbert des temps locaux browniens et une extension de la formule d'Itô (Brownian motion)

This paper is about the application to the function $(x-a)\log|x-a|-(x-a)$ (whose second derivative is $1/x-a$) of the Ito-Tanaka formula; the last term then involves a formal Hilbert transform $\tilde L^a_t$ of the local time process $L^a_t$. Such processes had been defined by Ito and McKean, and studied by Yamada as examples of Fukushima's ``additive functionals of zero energy''. Here it is proved, as a consequence of a general theorem, that this process has a jointly continuous version---more precisely, Hölder continuous of all orders $<1/2$ in $a$ and in $t$

Comment: For a modern version with references see Yor,

Keywords: Local times, Hilbert transform, Ito formula

Nature: Original

Retrieve article from Numdam

XVI: 22, 248-256, LNM 920 (1982)

Sur la convergence absolue de certaines intégrales (General theory of processes)

This paper is devoted to the a.s. absolute convergence of certain random integrals, a classical example of which is $\int_0^t ds/|B_s|^{\alpha}$ for Brownian motion starting from $0$. The author does not claim to prove deep results, but his technique of optional increasing reordering (réarrangement) of a process should be useful in other contexts too

Comment: This paper greatly simplifies a proof in the author's

Keywords: Enlargement of filtrations

Nature: Original

Retrieve article from Numdam

XVI: 23, 257-267, LNM 920 (1982)

Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T.~Chen (Stochastic calculus)

Consider a s.d.e. in a manifold, $dX_t=\sum_i A_i(X)\,dM^i_t$ (Stratonovich differentials), driven by continuous real semimartingales $M^i_t$, and where the $A_i$ have the geometrical nature of vector fields. Such an equation has a counterpart in which the $M^i(t)$ are arbitrary deterministic piecewise smooth functions, and if this equation can be solved by some deterministic machinery, then the s.d.e. can be solved too, just by making the input random. Thus a bridge is drawn between s.d.e.'s and problems of deterministic control theory. From this point of view, the complexity of the problem reflects that of the Lie algebra generated by the vector fields $A_i$. Assuming these fields are complete (i.e., generate true one-parameter groups on the manifold) and generate a finite dimensional Lie algebra (which then is the Lie algebra of a matrix group), the problem can be linearized. If the Lie algebra is nilpotent, the solution then can be expressed explicitly as a function of a finite number of iterated integrals of the driving processes (Chen integrals), and this provides the required ``deterministic machine''. It thus appears that results like those of Yamato (

Comment: See Kunita's paper 1432

Keywords: Stochastic differential equations, Lie algebras, Chen's iterated integrals, Campbell-Hausdorff formula

Nature: Original

Retrieve article from Numdam

XVI: 24, 268-284, LNM 920 (1982)

Sur le flot d'une équation différentielle stochastique (Stochastic calculus)

This paper is a companion to 1506, devoted to the main results on the flow of a (Lipschitz) stochastic differential equation driven by continous semimartingales: non-confluence of solutions from different initial points, surjectivity of the mapping, smooth dependence on the initial conditions. The proofs have been greatly simplified

Keywords: Stochastic differential equations, Flow of a s.d.e., Injectivity

Nature: Exposition, Original additions

Retrieve article from Numdam

XVI: 25, 285-297, LNM 920 (1982)

Un théorème de Helly pour les surmartingales fortes (Martingale theory)

Provide the set of (optional) strong supermartingales $X$ of the class (D) with the topology of weak $L^1$--convergence of $X_T$ at each stopping time $T$. Then it is shown that any subset which belongs uniformly to the class (D) is relatively compact, also in the sequential sense of extracting convergent subsequences

Comment: This paper was suggested by a similar result of Mokobodzki for strongly supermedian functions in potential theory

Keywords: Supermartingales, Strong supermartingales

Nature: Original

Retrieve article from Numdam

XVI: 26, 298-313, LNM 920 (1982)

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (General theory of processes)

This paper is a sequel to 1524. Let $\Theta$ be a

Keywords: Stopping times

Nature: Original

Retrieve article from Numdam

XVI: 27, 314-318, LNM 920 (1982)

Sur le théorème de la convergence dominée (General theory of processes, Stochastic calculus)

Consider previsible processes $U^n,U$ such that $U^n_T\rightarrow U_T$ in some sense at bounded previsible times $T$. The problem discussed is whether stochastic integrals $\int U^n_s dX_s$ converge (in the same sense) to $\int U_sdX_s$. Under a domination hypothesis, the answer is shown to be positive if the convergence is either weak convergence in $L^1$, or convergence in probability. The existence of the limiting process $U$ is not assumed in the paper; it is proved by a modification of an argument of Mokobodzki for which see 1110

Keywords: Stopping times, Optional processes, Weak convergence, Stochastic integrals

Nature: Original

Retrieve article from Numdam

XVI: 28, 319-337, LNM 920 (1982)

Sur la contiguïté de deux suites de mesures : généralisation d'un théorème de Kabanov-Liptser-Shiryayev

Retrieve article from Numdam

XVI: 29, 338-347, LNM 920 (1982)

À propos de l'intégrabilité uniforme des martingales exponentielles (Martingale theory)

Sufficient conditions are given for the uniform integrability of the exponential ${\cal E}(M)$, where $M$ is a local martingale with jumps $\ge-1$, refining older results of Lépingle and Mémin, and of the author. They involve the Lévy measure of the martingale

Comment: In the lemma p.339 delete the assumption $0<\beta$

Keywords: Exponential martingales

Nature: Original

Retrieve article from Numdam

XVI: 30, 348-354, LNM 920 (1982)

The total continuity of natural filtrations (General theory of processes)

Total continuity of a filtration ${\cal F}$ means that ${\cal F}_T={\cal F}_{T-}$ at every stopping time $T$, not necessarily previsible. It is shown that the filtration of a Lévy process without fixed discontinuities is totally continuous if and only if the jump size is a deterministic function of the jump time. Similarly, the natural filtration of a quasi-left continuous jump process is totally continuous if and only if the size of the $n$-th jump is a deterministic function of the jump times up to the $n$-th. It is shown that under the usual (here called ``strong'') previsible representation property, quasi-left continuity of the filtration implies total continuity

Keywords: Filtrations, Independent increments, Previsible representation, Total continuity, Lévy processes

Nature: Original

Retrieve article from Numdam

XVI: 31, 355-369, LNM 920 (1982)

Semimartingales à deux indices

Retrieve article from Numdam

XVI: 32, 370-379, LNM 920 (1982)

Semimartingales in predictable random open sets

Retrieve article from Numdam

XVI: 33, 380-383, LNM 920 (1982)

Intégrales stochastiques généralisées

Retrieve article from Numdam

XVI: 34, 384-391, LNM 920 (1982)

A.s. approximation results for multiplicative stochastic integrals

Retrieve article from Numdam

XVI: 35, 392-399, LNM 920 (1982)

There exists no ultimate solution to Skorokhod's problem

Retrieve article from Numdam

XVI: 36, 400-408, LNM 920 (1982)

Une propriété de domination de l'enveloppe de Snell des semimartingales fortes

Retrieve article from Numdam

XVI: 37, 409-411, LNM 920 (1982)

Une remarque sur l'approximation des solutions d'e.d.s.

Retrieve article from Numdam

XVI: 38, 412-441, LNM 920 (1982)

On some limit theorems for solutions of stochastic differential equations

Retrieve article from Numdam

XVI: 39, 442-446, LNM 920 (1982)

Équations différentielles linéaires : la méthode de variation des constantes

Retrieve article from Numdam

XVI: 40, 447-450, LNM 920 (1982)

Quelques remarques sur un nouveau type d'équations différentielles stochastiques

Retrieve article from Numdam

XVI: 41, 451-468, LNM 920 (1982)

Stochastic differential equations with feedback in the differentials

Retrieve article from Numdam

XVI: 42, 469-489, LNM 920 (1982)

Règle maximale

Retrieve article from Numdam

XVI: 43, 490-502, LNM 920 (1982)

Pathwise differentiability with respect to a parameter of solutions of stochastic differential equations

Retrieve article from Numdam

XVI: 44, 503-508, LNM 920 (1982)

Résultats d'Atkinson sur les processus de Markov

Retrieve article from Numdam

XVI: 45, 509-514, LNM 920 (1982)

Hypothesis (B) of Hunt

Retrieve article from Numdam

XVI: 46, 515-518, LNM 920 (1982)

An extension of Motoo's theorem

Retrieve article from Numdam

XVI: 47, 519-543, LNM 920 (1982)

An integral representation of randomized probabilities and its applications

Retrieve article from Numdam

XVI: 48, 544-569, LNM 920 (1982)

Topologies métrisables rendant continues les trajectoires d'un processus

Retrieve article from Numdam

XVI: 49, 570-580, LNM 920 (1982)

Mesures gaussiennes et mesures produits

Retrieve article from Numdam

XVI: 50, 581-601, LNM 920 (1982)

Sur la densité du maximum d'une fonction aléatoire gaussienne

Retrieve article from Numdam

XVI: 51, 602-608, LNM 920 (1982)

Sur la loi du logarithme itéré dans les espaces réflexifs

Retrieve article from Numdam

XVI: 52, 609-622, LNM 920 (1982)

La loi du logarithme itéré pour les variables aléatoires prégaussiennes à valeurs dans un espace de Banach à norme régulière

Retrieve article from Numdam

XVI: 53, 623-623, LNM 920 (1982)

Correction au Séminaire XV

Retrieve article from Numdam

XVI: 54, 623-623, LNM 920 (1982)

Addendum au Séminaire XV

Retrieve article from Numdam