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XV: 01, 1-5, LNM 850 (1981)

**FERNIQUE, Xavier**

Sur les lois de certaines intégrales associées à des mouvements browniens (Brownian motion)

Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)

Keywords: Iterated stochastic integrals

Nature: Original

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XV: 02, 6-10, LNM 850 (1981)

**FERNIQUE, Xavier**

Sur le théorème de Kantorovitch-Rubinstein dans les espaces polonais (Measure theory)

The theorem asserts the existence, given two probability measures $\mu,\nu$ on a complete separable metric space $(S,d)$, of a measure $\pi$ on $S\times S$ with marginals $\mu$ and $\nu$ such that $\int d(x,y)\,\pi(dx,dy)$ realizes a suitable distance between $\mu$ and $\nu$. An elementary proof is given here by reduction to the compact case

Keywords: Convergence in law

Nature: New proof of known results

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XV: 03, 11-37, LNM 850 (1981)

**LEDOUX, Michel**

La loi du logarithme itéré bornée dans les espaces de Banach (Banach space valued random variables)

To be completed

Keywords: Law of the iterated logarithm

Nature: Original

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XV: 04, 38-43, LNM 850 (1981)

**NOBELIS, Photis**

Fonctions aléatoires lipschitziennes (Regularity of random processes)

A sufficient condition is given so that almost all sample functions of a random process defined on $[0,1]^**N**$ satisfy a Lipschitz condition (involving a general modulus of continuity). The method is that of majorizing measures. A condition due to Ibragimov is extended

Keywords: Majorizing measures

Nature: Original

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XV: 05, 44-102, LNM 850 (1981)

**MEYER, Paul-André**

Géométrie stochastique sans larmes (Stochastic differential geometry)

Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is*continuous semimartingales in manifolds,* following L.~Schwartz (LN **780**, 1980), but with additional features: an indication of J.M.~Bismut hinting to a definition of continuous *martingales * in a manifold, and the author's own interest on the forgotten intrinsic definition of the second differential $d^2f$ of a function. All this fits together into a geometric approach to semimartingales, and a probabilistic approach to such geometric topics as torsion-free connexions

Comment: A short introduction by the same author can be found in*Stochastic Integrals,* Springer LNM 851. The same ideas are expanded and presented in the supplement to Volume XVI and the book by Émery, *Stochastic Calculus on Manifolds *

Keywords: Semimartingales in manifolds, Martingales in manifolds, Transfer principle, Stochastic differential equations, Stochastic integrals, Stratonovich integrals

Nature: Original

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XV: 06, 103-117, LNM 850 (1981)

**MEYER, Paul-André**

Flot d'une équation différentielle stochastique (Stochastic calculus)

Malliavin showed very neatly how an (Ito) stochastic differential equation on $**R**^n$ with $C^{\infty}$ coefficients, driven by Brownian motion, generates a flow of diffeomorphisms. This consists of three results: smoothness of the solution as a function of its initial point, showing that the mapping is 1--1, and showing that it is onto. The last point is the most delicate. Here the results are extended to stochastic differential equations on $**R**^n$ driven by continuous semimartingales, and only partially to the case of semimartingales with jumps. The essential argument is borrowed from Kunita and Varadhan (see Kunita's talk in the Proceedings of the Durham Symposium on SDE's, LN 851)

Comment: The results on semimartingales with jumps have been proved independently by Uppman. Some dust has been swept under the rugs about the non-explosion of the solution, and the results should be considered valid only in the globally Lipschitz case. See also Uppman 1624 and Léandre 1922

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

Nature: Exposition, Original additions

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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 transport

Nature: Original

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XV: 08, 142-142, LNM 850 (1981)

**MEYER, Paul-André**

Une question de théorie des processus (Stochastic calculus)

It is remarked that the stochastic integrals that appear in stochastic differential geometry are of a particular kind, and asked whether the theory could be developed for processes belonging to a larger class than semimartingales

Comment: For recent work in this area, see T. Lyons' article in*Rev. Math. Iberoamericana* 14 (1998) on differential equations driven by non-smooth functions

Keywords: Semimartingales

Nature: Open question

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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 variation

Nature: Original

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XV: 10, 151-166, LNM 850 (1981)

**MEYER, Paul-André**

Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)

The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (*J. Funct. Anal.*, 38, 1980)

Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ

Nature: Original

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XV: 11, 167-188, LNM 850 (1981)

**BOULEAU, Nicolas**

Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov (Markov processes)

The main result of the paper of Kunita (*Nagoya Math. J.*, **36**, 1969) showed that the domain of the extended generator $A$ of a right Markov semigroup is an algebra if and only if the angle brackets of all martingales are absolutely continuous with respect to the measure $dt$. See also 1010. Such semigroups are called here ``semigroups of Lebesgue type''. Kunita's result is sharpened here: it is proved in particular that if some non-affine convex function $f$ operates on the domain, then the semigroup is of Lebesgue type (Kunita's result corresponds to $f(x)=x^2$) and if the second derivative of $f$ is not absolutely continuous, then the semigroup has no diffusion part (i.e., all martingales are purely discontinuous). The second part of the paper is devoted to the behaviour of the extended domain under an absolutely continuous change of probability (arising from a multiplicative functional)

Keywords: Semigroup theory, Carré du champ, Infinitesimal generators

Nature: Original

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XV: 12, 189-190, LNM 850 (1981)

**BARLOW, Martin T.**

On Brownian local time (Brownian motion)

Let $(L^a_t)$ be the standard (jointly continuous) version of Brownian local times. Perkins has shown that for fixed $t$ $a\mapsto L^a_t$ is a semimartingale relative to the excusrion fields. An example is given of a stopping time $T$ for which $a\mapsto L^a_T$ is not a semimartingale

Keywords: Local times

Nature: Original

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XV: 13, 191-205, LNM 850 (1981)

**MAISONNEUVE, Bernard**

On Lévy's downcrossing theorem and various extensions (Excursion theory)

Lévy's downcrossing theorem describes the local time $L_t$ of Brownian motion at $0$ as the limit of $\epsilon D_t(\epsilon)$, where $D_t$ denotes the number of downcrossings of the interval $(0,\epsilon)$ up to time $t$. To give a simple proof of this result from excursion theory, easy to generalize, the paper uses the weaker definition of regenerative systems described in 1137. A gap in the related author's paper*Zeit. für W-Theorie,* **52**, 1980 is repaired at the end of the paper

Keywords: Excursions, Lévy's downcrossing theorem, Local times, Regenerative systems

Nature: Original

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XV: 14, 206-209, LNM 850 (1981)

**McGILL, Paul**

A direct proof of the Ray-Knight theorem (Brownian motion)

The (first) Ray-Knight theorem describes the law of the process $(L_T^{1-a})_{0\le a\le 1}$ where $(L^a_t)$ is the family of local times of Brownian motion starting from $0$ and $T$ is the hitting time of $1$. A direct proof is given indeed. It is reproduced in Revuz-Yor,*Continuous Martingales and Brownian Motion,* Chapter XI, exercice (2.7)

Keywords: Local times, Ray-Knight theorems, Bessel processes

Nature: New proof of known results

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XV: 15, 210-226, LNM 850 (1981)

**JEULIN, Thierry**; **YOR, Marc**

Sur les distributions de certaines fonctionnelles du mouvement brownien (Brownian motion)

This paper gives new proofs and extensions of results due to Knight, concerning occupation times by the process $(S_t,B_t)$ up to time $T_a$, where $(B_t)$ is Brownian motion, $T_a$ the hitting time of $a$, and $(S_t)$ is $\sup_{s\le t} B_s$. The method uses enlargement of filtrations, and martingales similar to those of 1306. Theorem 3.7 is a decomposition of Brownian paths akin to Williams' decomposition

Comment: See also 1516

Keywords: Explicit laws, Occupation times, Enlargement of filtrations, Williams decomposition

Nature: Original

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XV: 16, 227-250, LNM 850 (1981)

**ROGERS, L.C.G.**

Williams' characterization of the Brownian excursion law: proof and applications (Brownian motion)

In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams*Diffusions, Markov Processes and Martingales,* Wiley 1979, II.67), and it collects a number of applications, including the Azéma-Yor approach to Skorohod's imbedding theorem (1306)

Keywords: Excursions, Explicit laws, Bessel processes, Skorohod imbedding

Nature: Original

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XV: 17, 251-258, LNM 850 (1981)

**PITMAN, James W.**

A note on $L_2$ maximal inequalities (Martingale theory)

This paper contains a $L^2$ inequality between two processes $(X_n,M_n)$ under assumptions which (if $X$ is a martingale) apply to $M_n=\sup_{m\le n} |X_m|$, and to other interesting cases as well. In particular, Doob's inequality is valid for the larger process $\sup_{m\le n} X_m^+ +\sup_{m\le n} X_m^-$

Keywords: Maximal inequality, Doob's inequality

Nature: Original

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XV: 18, 259-277, LNM 850 (1981)

**BRU, Bernard**; **HEINICH, Henri**; **LOOTGIETER, Jean-Claude**

Autour de la dualité $(H^1,BMO)$ (Martingale theory)

This is a sequel to 1330. Given two martingales $(X,Y)$ in $H^1$ and $BMO$, it is investigated whether their duality functional can be safely estimated as $E[X_{\infty}Y_{\infty}]$. The simple result is that if $X_{\infty}Y_{\infty}$ belongs to $L^1$, or merely is bounded upwards by an element of $L^1$, then the answer is positive. The second (and longer) part of the paper searches for subspaces of $H^1$ and $BMO$ such that the property would hold between their elements, and here the results are fragmentary (a question of 1330 is answered). An appendix discusses a result of Talagrand

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

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XV: 19, 278-284, LNM 850 (1981)

**ÉMERY, Michel**

Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)

Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential

Keywords: $BMO$

Nature: Exposition, Original additions

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XV: 20, 285-289, LNM 850 (1981)

**CHOU, Ching Sung**

Une inégalité de martingales avec poids (Martingale theory)

Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)

Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions

Nature: Original

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XV: 21, 290-306, LNM 850 (1981)

**CHACON, Rafael V.**; **LE JAN, Yves**; **WALSH, John B.**

Spatial trajectories (Markov processes, General theory of processes)

It is well known that Markov processes with the same excessive functions are the same up to a strictly increasing continuous time-change. It is therefore natural to study spatial trajectories, i.e., trajectories up to a strictly increasing continuous time changes, and in particular to provide the space of all spatial trajectories with a reasonable $\sigma$-field so that it may carry measures. It is shown here that the space of right-continuous spatial trajectories with left-hand limits is a Blackwell space. The class of intrinsic stopping times defined on this space is also investigated

Comment: See Chacon-Jamison,*Israel J. of M.*, **33**, 1979

Keywords: Spatial trajectories

Nature: Original

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XV: 22, 307-310, LNM 850 (1981)

**LE JAN, Yves**

Tribus markoviennes et prédiction (Markov processes, General theory of processes)

The problem discussed here is whether a given filtration is generated by a Ray process. The answer is positive under very general conditions. Knight's prediction theory (1007) is used

Keywords: Prediction theory

Nature: Original

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XV: 23, 311-319, LNM 850 (1981)

**ALDOUS, David J.**; **BARLOW, Martin T.**

On countable dense random sets (General theory of processes, Point processes)

This paper is devoted to random sets $B$ which are countable, optional (i.e., can be represented as the union of countably many graphs of stopping times $T_n$) and dense. The main result is that whenever the increasing processes $I_{t\ge T_n}$ have absolutely continuous compensators (in which case the same property holds for any stopping time $T$ whose graph is contained in $B$), then the random set $B$ can be represented as the union of all the points of countably many independent standard Poisson processes (intuitively, a Poisson measure whose rate is $+\infty$ times Lebesgue measure). This may require, however, an innocuous enlargement of filtration. Another characterization of such random sets is roughly that they do not intersect previsible sets of zero Lebesgue measure. Note also an interesting example of a set optional w.r.t. two filtrations, but not w.r.t. their intersection

Keywords: Poisson point processes

Nature: Original

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XV: 24, 320-346, LNM 850 (1981)

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

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

The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in 1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)

Keywords: General filtrations, Strong supermartingales, Snell's envelope, Section theorems

Nature: Original

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XV: 25, 347-350, LNM 850 (1981)

**MAISONNEUVE, Bernard**

Surmartingales-mesures (Martingale theory)

Consider a discrete filtration $({\cal F}_n)$ and let ${\cal A}$ be the algebra, $\cup_n {\cal F}_n$, generating a $\sigma$-algebra ${\cal F}_\infty$. A positive supermartingale $(X_n)$ is called a supermartingale measure if the set function $A\mapsto\lim_n\int_A X_n\,dP$ on $A$ is $\sigma$-additive, and thus can be extended to a measure $\mu$. Then the Lebesgue decomposition of this measure is described (theorem 1). More generally, the Lebesgue decomposition of any measure $\mu$ on ${\cal F}_\infty$ is described. This is meant to complete theorem III.1.5 in Neveu,*Martingales à temps discret *

Comment: The author points out at the end that theorem 2 had been already proved by Horowitz (*Zeit. für W-theorie,* 1978) in continuous time. This topic is now called Kunita decomposition, see 1005 and the corresponding references

Keywords: Supermartingales, Kunita decomposition

Nature: Original

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XV: 26, 351-370, LNM 850 (1981)

**DELLACHERIE, Claude**

Mesurabilité des débuts et théorème de section~: le lot à la portée de toutes les bourses (General theory of processes)

One of the main topics in these seminars has been the application to stochastic processes of results from descriptive set theory and capacity theory, at different levels. Since these results are considered difficult, many attempts have been made to shorten and simplify the exposition. A noteworthy one was 511, in which Dellacherie introduced ``rabotages'' (306) to develop the theory without analytic sets; see also 1246, 1255. The main feature of this paper is a new interpretation of rabotages as a two-persons game, ascribed to Telgarsky though no reference is given, leading to a pleasant exposition of the whole theory and its main applications

Keywords: Section theorems, Capacities, Sierpinski's ``rabotages''

Nature: Original

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XV: 27, 371-387, LNM 850 (1981)

**DELLACHERIE, Claude**

Sur les noyaux $\sigma$-finis (Measure theory)

This paper is an improvement of 1235. Assume $(X,{\cal X})$ and $(Y,{\cal Y})$ are measurable spaces and $m(x,A)$ is a kernel, i.e., is measurable in $x\in X$ for $A\in{\cal Y}$, and is a $\sigma$-finite measure in $A$ for $x\in X$. Then the problem is to represent the measures $m(x,dy)$ as $g(x,y)\,N(x,dy)$ where $g$ is a jointly measurable function and $N$ is a Markov kernel---possibly enlarging the $\sigma$-field ${\cal X}$ to include analytic sets. The crucial hypothesis (called*measurability * of $m$) is the following: for every auxiliary space $(Z, {\cal Z})$, the mapping $(x,z)\mapsto m_x\otimes \epsilon_z$ is again a kernel (in fact, the auxiliary space $**R**$ is all one needs). The case of ``basic'' kernels, considered in 1235, is thoroughly discussed

Keywords: Kernels, Radon-Nikodym theorem

Nature: Original

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XV: 28, 388-398, LNM 850 (1981)

**SPILIOTIS, Jean**

Sur les travaux de Krylov en théorie de l'intégrale stochastique (Martingale theory)

The well-known work of Malliavin deals with the existence of smooth densities for solutions of stochastic differential equations with smooth coefficients satisfying a hypoellipticity condition. N.V.~Krylov's earlier work (among many papers see*Izvestija Akad Nauk,* **38**, 1974, and Krylov's book *Controlled Diffusion processes,* Springer 1980) dealt with the existence of densities for several dimensional stochastic integrals with measurable bounded integrands, satisfying an ellipticity condition. It is a puzzling fact that nobody ever succeeded in unifying these results. Krylov's method depends on results of the Russian school on Monge-Ampère equations (see Pogorelov *The Minkowski Multidimensional Problem,* 1978). This exposition attempts, rather modestly, to explain in the seminar's language what it is all about, and in particular to show the place where a crucial lemma on convex functions is used

Keywords: Stochastic integrals, Existence of densities

Nature: Exposition

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XV: 29, 399-412, LNM 850 (1981)

**YOEURP, Chantha**

Sur la dérivation stochastique au sens de Davis (Stochastic calculus, Brownian motion)

The problem is that of estimating $H_t$ from a knowledge of the process $\int H_sdX_s$, where $X_t$ is a continuous (semi)martingale, as a limit of ratios of the form $\int_t^{t+\alpha} H_sdX_s/(X_{t+\alpha}-X_t)$, or replacing $t+\alpha$ by suitable stopping times

Comment: The problem was suggested and partially solved by Mark H.A. Davis (*Teor. Ver. Prim.*, **20**, 1975, 887--892). See also 1428

Keywords: Stochastic integrals

Nature: Original

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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 integrals

Nature: Original

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XV: 31, 490-492, LNM 850 (1981)

**STRICKER, Christophe**

Sur deux questions posées par Schwartz (Stochastic calculus)

Schwartz studied semimartingales in random open sets, and raised two questions: Given a semimartingale $X$ and a random open set $A$, 1) Assume $X$ is increasing in every subinterval of $A$; then is $X$ equal on $A$ to an increasing adapted process on the whole line? 2) Same statement with ``increasing'' replaced by ``continuous''. Schwartz could prove statement 1) assuming $X$ was continuous. It is proved here that 1) is false if $X$ is only cadlag, and that 2) is false in general, though it is true if $A$ is previsible, or only accessible

Keywords: Random sets, Semimartingales in a random open set

Nature: Original

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XV: 32, 493-498, LNM 850 (1981)

**STRICKER, Christophe**

Quasi-martingales et variations (Martingale theory)

This paper contains remarks on quasimartingales, the most useful of which being perhaps the fact that, for a right-continuous process, the stochastic variation is the same with respect to the filtrations $({\cal F}_{t})$ and $({\cal F}_{t-})$

Keywords: Quasimartingales

Nature: Original

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XV: 33, 499-522, LNM 850 (1981)

**STRICKER, Christophe**

Quelques remarques sur la topologie des semimartingales. Applications aux intégrales stochastiques (Stochastic calculus)

This paper contains a number of useful technical results on the topology of semimartingales (see 1324), some of which were previously known with more complicated proofs. In particular, it is shown how to improve the convergence of sequences of semimartingales by a convenient change of probability. The topology of semimartingales is used to handle elegantly the stochastic integration of previsible processes which are not locally bounded (see 1415). Finally, boundedness of a set of semimartingales is shown to be equivalent to the boundedness (in an elementary sense) of a set of increasing processes controlling them in the sense of Métivier-Pellaumail (see 1412, 1413, 1414)

Keywords: Semimartingales, Stochastic integrals, Spaces of semimartingales, Métivier-Pellaumail inequality

Nature: Original

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XV: 34, 523-525, LNM 850 (1981)

**STRICKER, Christophe**

Sur la caractérisation des semi-martingales (General theory of processes, Stochastic calculus)

This is a sequel to the preceding paper 1533, giving a simple proof that any semimartingale may be brought into any class ${\cal S}^p$ by a convenient change of probability

Keywords: Semimartingales, Spaces of semimartingales

Nature: Original

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XV: 35, 526-528, LNM 850 (1981)

**YOR, Marc**

Sur certains commutateurs d'une filtration (General theory of processes)

Let $({\cal F}_t)$ be a filtration satisfying the usual conditions and ${\cal G}$ be a $\sigma$-field. Then the conditional expectation $E[.|{\cal G}]$ commutes with $E[.|{\cal F}_T]$ for all stopping times $T$ if and only if for some stopping time $S$ ${\cal G}$ lies between ${\cal F}_{S-}]$ and ${\cal F}_S]$

Keywords: Conditional expectations

Nature: Original

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XV: 36, 529-546, LNM 850 (1981)

**JACOD, Jean**; **MÉMIN, Jean**

Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (Measure theory)

For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $**R**$. Let us define a fuzzy r.v. $X$ on $(\Omega, {\cal F},P)$ as a probability measure on $\Omega\times**R**$ whose projection on $\Omega$ is $P$. In particular, a standard r.v. $X$ defines such a measure as the image of $P$ under the map $\omega\mapsto (\omega,X(\omega))$. The space of fuzzy r.v.'s is provided with a weak topology, associated with the bounded functions $f(\omega,x)$ which are continuous in $x$ for every $\omega$, or equivalently with the functions $I_A(\omega)\,f(x)$ with $f$ bounded continuous. The main topic of this paper is the study of this topology

Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see 1228

Keywords: Fuzzy random variables, Convergence in law

Nature: Original

Retrieve article from Numdam

XV: 37, 547-560, LNM 850 (1981)

**JACOD, Jean**

Convergence en loi de semimartingales et variation quadratique (General theory of processes, Stochastic calculus)

The convergence in law of cadlag processes to a cadlag process being understood in the sense of Skorohod, the problem is to find sufficient conditions under which, given semimartingales $X^n$ and $X$ such that $X^n\rightarrow X$ in law, one may deduce that $[X^n,X^n]$ converges in law to $[X,X]$. This is achieved assuming a uniform bound on the expectations of the supremum of the jumps. A version of the theorem applied to processes which are not semimartingales, but are equal to semimartingales on large sets

Keywords: Semimartingales, Skorohod topology, Convergence in law

Nature: Original

Retrieve article from Numdam

XV: 38, 561-586, LNM 850 (1981)

**PELLAUMAIL, Jean**

Solutions faibles et semi-martingales (Stochastic calculus, General theory of processes)

From the author's summary: ``we consider a stochastic differential equation $dX=a(X)\,dZ$ where $Z$ is a semimartingale and $a$ is a previsible functional which is continuous for the uniform norm. We prove the existence of a weak solution for such an equation''. The important point is the definition of a weak solution: it turns out to be a ``fuzzy process'' in the sense of 1536, i.e., a fuzzy r.v. taking values in the Polish space of cadlag sample functions

Keywords: Stochastic differential equations, Weak solutions, Fuzzy random variables

Nature: Original

Retrieve article from Numdam

XV: 39, 587-589, LNM 850 (1981)

**ÉMERY, Michel**

Non-confluence des solutions d'une équation stochastique lipschitzienne (Stochastic calculus)

This paper proves that the solutions of a stochastic differential equation $dX_t=f(., t,X_t)\,dM_t$ driven by a continuous semimartingale $M$, where $f(\omega,t,x)$ is as usual previsible in $\omega$ and Lipschitz in $x$, are non-confluent, i.e., the solutions starting at different points never meet

Comment: See also 1506, 1507 (for less general s.d.e.'s), and 1624

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

Nature: Original

Retrieve article from Numdam

XV: 40, 590-603, LNM 850 (1981)

**STROOCK, Daniel W.**; **YOR, Marc**

Some remarkable martingales (Martingale theory)

This is a sequel to a well-known paper by the authors (*Ann. ENS,* **13**, 1980) on the subject of pure martingales. A continuous martingale $(M_t)$ with $<M,M>_{\infty}=\infty$ is pure if the time change which reduces it to a Brownian motion $(B_t)$ entails no loss of information, i.e., if $M$ is measurable w.r.t. the $\sigma$-field generated by $B$. The first part shows the purity of certain stochastic integrals. Among the striking examples considered, the stochastic integrals $\int_0^t B^n_sdB_s$ are extremal for every integer $n$, pure for $n$ odd, but nothing is known for $n$ even. A beautiful result unrelated to purity is the following: complex Brownian motion $Z_t$ starting at $z_0$ and its (Lévy) area integral generate the same filtration if and only if $z_0\neq0$

Keywords: Pure martingales, Previsible representation

Nature: Original

Retrieve article from Numdam

XV: 41, 604-617, LNM 850 (1981)

**LÉPINGLE, Dominique**; **MEYER, Paul-André**; **YOR, Marc**

Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)

This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case

Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps

Nature: Original

Retrieve article from Numdam

XV: 42, 618-626, LNM 850 (1981)

**ITMI, Mhamed**

Processus ponctuels marqués stochastiques. Représentation des martingales et filtration naturelle quasicontinue à gauche (General theory of processes)

This paper contains a study of the filtration generated by a point process (multivariate: it takes values in a Polish space), and in particular of its quasi-left continuity, and previsible representation

Keywords: Point processes, Previsible representation

Nature: Original

Retrieve article from Numdam

XV: 43, 627-631, LNM 850 (1981)

**WANG, Jia-Gang**

Some remarks on processes with independent increments (Independent increments)

This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also 1544

Keywords: Poisson processes, Lévy measures

Nature: Original

Retrieve article from Numdam

XV: 44, 632-642, LNM 850 (1981)

**SIDIBÉ, Ramatoulaye**

Mesures à accroissements indépendants et P.A.I. non homogènes (Independent increments)

This is an improved version of 1310: the classical theorem of Lévy on the structure of processes with independent increments is elegantly proved by martingale methods, in the non-homogeneous case, and it is proved that the process is a special semimartingale if and only if it is integrable

Nature: New proof of known results

Retrieve article from Numdam

XV: 45, 643-668, LNM 850 (1981)

**AUERHAN, J.**; **LÉPINGLE, Dominique**

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (General theory of processes, Brownian motion, Martingale theory)

This is a sequel to 1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in 1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given

Comment: Further results are given by Malric*Ann. Inst. H. Poincaré * **26** (1990)

Nature: Original

Retrieve article from Numdam

XV: 46, 669-670, LNM 850 (1981)

**LÉPINGLE, Dominique**

Une remarque sur les lois de certains temps d'atteinte (Brownian motion)

Let $T$ be the exit time of the interval $[-d,c]$ for a Brownian motion starting at $0$. A classical formula giving the Laplace transform of the law of $T$ can be extended by analytical continuation to the positive axis. It is shown here that this extension has a purely probabilistic proof. The same method gives two other formulas

Keywords: Exit time from an interval

Nature: New proof of known results

Retrieve article from Numdam

XV: 47, 671-672, LNM 850 (1981)

**BAKRY, Dominique**

Une remarque sur les semi-martingales à deux indices (Several parameter processes)

Let $({\cal F}^1_s)$ and $({\cal F}^2_t)$ be two filtrations whose conditional expectations commute. Let $(A_t)$ be a bounded increasing process adapted to $({\cal F}^2_t)$. It had been proved under stringent absolute continuity conditions on $A$ that the process $X_{st}=E[A_t\,|\,{\cal F}^1_s]$ was a semimartingale (a stochastic integrator). A counterexample is given here to show that this is not true in general

Keywords: Two-parameter semimartingales

Nature: Original

Retrieve article from Numdam

XV: 48, 673-688, LNM 850 (1981)

**MAZZIOTTO, Gérald**; **SZPIRGLAS, Jacques**

Un exemple de processus à deux indices sans l'hypothèse F4 (Several parameter processes)

A natural two-parameter filtration is associated with a random point in the positive quadrant $**R**^2_+$. Though it does not satisfy in general the Cairoli-Walsh commutation property called F4, it is possible to develop for this filtration a reasonable theory of optional and previsible processes, projection theorems, etc

Keywords: Two-parameter processes

Nature: Original

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Sur les lois de certaines intégrales associées à des mouvements browniens (Brownian motion)

Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)

Keywords: Iterated stochastic integrals

Nature: Original

Retrieve article from Numdam

XV: 02, 6-10, LNM 850 (1981)

Sur le théorème de Kantorovitch-Rubinstein dans les espaces polonais (Measure theory)

The theorem asserts the existence, given two probability measures $\mu,\nu$ on a complete separable metric space $(S,d)$, of a measure $\pi$ on $S\times S$ with marginals $\mu$ and $\nu$ such that $\int d(x,y)\,\pi(dx,dy)$ realizes a suitable distance between $\mu$ and $\nu$. An elementary proof is given here by reduction to the compact case

Keywords: Convergence in law

Nature: New proof of known results

Retrieve article from Numdam

XV: 03, 11-37, LNM 850 (1981)

La loi du logarithme itéré bornée dans les espaces de Banach (Banach space valued random variables)

To be completed

Keywords: Law of the iterated logarithm

Nature: Original

Retrieve article from Numdam

XV: 04, 38-43, LNM 850 (1981)

Fonctions aléatoires lipschitziennes (Regularity of random processes)

A sufficient condition is given so that almost all sample functions of a random process defined on $[0,1]^

Keywords: Majorizing measures

Nature: Original

Retrieve article from Numdam

XV: 05, 44-102, LNM 850 (1981)

Géométrie stochastique sans larmes (Stochastic differential geometry)

Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is

Comment: A short introduction by the same author can be found in

Keywords: Semimartingales in manifolds, Martingales in manifolds, Transfer principle, Stochastic differential equations, Stochastic integrals, Stratonovich integrals

Nature: Original

Retrieve article from Numdam

XV: 06, 103-117, LNM 850 (1981)

Flot d'une équation différentielle stochastique (Stochastic calculus)

Malliavin showed very neatly how an (Ito) stochastic differential equation on $

Comment: The results on semimartingales with jumps have been proved independently by Uppman. Some dust has been swept under the rugs about the non-explosion of the solution, and the results should be considered valid only in the globally Lipschitz case. See also Uppman 1624 and Léandre 1922

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

Nature: Exposition, Original additions

Retrieve article from Numdam

XV: 07, 118-141, LNM 850 (1981)

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

Keywords: Stochastic differential equations, Flow of a s.d.e., Change of variable formula, Stochastic parallel transport

Nature: Original

Retrieve article from Numdam

XV: 08, 142-142, LNM 850 (1981)

Une question de théorie des processus (Stochastic calculus)

It is remarked that the stochastic integrals that appear in stochastic differential geometry are of a particular kind, and asked whether the theory could be developed for processes belonging to a larger class than semimartingales

Comment: For recent work in this area, see T. Lyons' article in

Keywords: Semimartingales

Nature: Open question

Retrieve article from Numdam

XV: 09, 143-150, LNM 850 (1981)

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

Keywords: Stochastic integrals, Change of variable formula, Quadratic variation

Nature: Original

Retrieve article from Numdam

XV: 10, 151-166, LNM 850 (1981)

Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)

The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (

Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ

Nature: Original

Retrieve article from Numdam

XV: 11, 167-188, LNM 850 (1981)

Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov (Markov processes)

The main result of the paper of Kunita (

Keywords: Semigroup theory, Carré du champ, Infinitesimal generators

Nature: Original

Retrieve article from Numdam

XV: 12, 189-190, LNM 850 (1981)

On Brownian local time (Brownian motion)

Let $(L^a_t)$ be the standard (jointly continuous) version of Brownian local times. Perkins has shown that for fixed $t$ $a\mapsto L^a_t$ is a semimartingale relative to the excusrion fields. An example is given of a stopping time $T$ for which $a\mapsto L^a_T$ is not a semimartingale

Keywords: Local times

Nature: Original

Retrieve article from Numdam

XV: 13, 191-205, LNM 850 (1981)

On Lévy's downcrossing theorem and various extensions (Excursion theory)

Lévy's downcrossing theorem describes the local time $L_t$ of Brownian motion at $0$ as the limit of $\epsilon D_t(\epsilon)$, where $D_t$ denotes the number of downcrossings of the interval $(0,\epsilon)$ up to time $t$. To give a simple proof of this result from excursion theory, easy to generalize, the paper uses the weaker definition of regenerative systems described in 1137. A gap in the related author's paper

Keywords: Excursions, Lévy's downcrossing theorem, Local times, Regenerative systems

Nature: Original

Retrieve article from Numdam

XV: 14, 206-209, LNM 850 (1981)

A direct proof of the Ray-Knight theorem (Brownian motion)

The (first) Ray-Knight theorem describes the law of the process $(L_T^{1-a})_{0\le a\le 1}$ where $(L^a_t)$ is the family of local times of Brownian motion starting from $0$ and $T$ is the hitting time of $1$. A direct proof is given indeed. It is reproduced in Revuz-Yor,

Keywords: Local times, Ray-Knight theorems, Bessel processes

Nature: New proof of known results

Retrieve article from Numdam

XV: 15, 210-226, LNM 850 (1981)

Sur les distributions de certaines fonctionnelles du mouvement brownien (Brownian motion)

This paper gives new proofs and extensions of results due to Knight, concerning occupation times by the process $(S_t,B_t)$ up to time $T_a$, where $(B_t)$ is Brownian motion, $T_a$ the hitting time of $a$, and $(S_t)$ is $\sup_{s\le t} B_s$. The method uses enlargement of filtrations, and martingales similar to those of 1306. Theorem 3.7 is a decomposition of Brownian paths akin to Williams' decomposition

Comment: See also 1516

Keywords: Explicit laws, Occupation times, Enlargement of filtrations, Williams decomposition

Nature: Original

Retrieve article from Numdam

XV: 16, 227-250, LNM 850 (1981)

Williams' characterization of the Brownian excursion law: proof and applications (Brownian motion)

In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams

Keywords: Excursions, Explicit laws, Bessel processes, Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XV: 17, 251-258, LNM 850 (1981)

A note on $L_2$ maximal inequalities (Martingale theory)

This paper contains a $L^2$ inequality between two processes $(X_n,M_n)$ under assumptions which (if $X$ is a martingale) apply to $M_n=\sup_{m\le n} |X_m|$, and to other interesting cases as well. In particular, Doob's inequality is valid for the larger process $\sup_{m\le n} X_m^+ +\sup_{m\le n} X_m^-$

Keywords: Maximal inequality, Doob's inequality

Nature: Original

Retrieve article from Numdam

XV: 18, 259-277, LNM 850 (1981)

Autour de la dualité $(H^1,BMO)$ (Martingale theory)

This is a sequel to 1330. Given two martingales $(X,Y)$ in $H^1$ and $BMO$, it is investigated whether their duality functional can be safely estimated as $E[X_{\infty}Y_{\infty}]$. The simple result is that if $X_{\infty}Y_{\infty}$ belongs to $L^1$, or merely is bounded upwards by an element of $L^1$, then the answer is positive. The second (and longer) part of the paper searches for subspaces of $H^1$ and $BMO$ such that the property would hold between their elements, and here the results are fragmentary (a question of 1330 is answered). An appendix discusses a result of Talagrand

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

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XV: 19, 278-284, LNM 850 (1981)

Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)

Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential

Keywords: $BMO$

Nature: Exposition, Original additions

Retrieve article from Numdam

XV: 20, 285-289, LNM 850 (1981)

Une inégalité de martingales avec poids (Martingale theory)

Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)

Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions

Nature: Original

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XV: 21, 290-306, LNM 850 (1981)

Spatial trajectories (Markov processes, General theory of processes)

It is well known that Markov processes with the same excessive functions are the same up to a strictly increasing continuous time-change. It is therefore natural to study spatial trajectories, i.e., trajectories up to a strictly increasing continuous time changes, and in particular to provide the space of all spatial trajectories with a reasonable $\sigma$-field so that it may carry measures. It is shown here that the space of right-continuous spatial trajectories with left-hand limits is a Blackwell space. The class of intrinsic stopping times defined on this space is also investigated

Comment: See Chacon-Jamison,

Keywords: Spatial trajectories

Nature: Original

Retrieve article from Numdam

XV: 22, 307-310, LNM 850 (1981)

Tribus markoviennes et prédiction (Markov processes, General theory of processes)

The problem discussed here is whether a given filtration is generated by a Ray process. The answer is positive under very general conditions. Knight's prediction theory (1007) is used

Keywords: Prediction theory

Nature: Original

Retrieve article from Numdam

XV: 23, 311-319, LNM 850 (1981)

On countable dense random sets (General theory of processes, Point processes)

This paper is devoted to random sets $B$ which are countable, optional (i.e., can be represented as the union of countably many graphs of stopping times $T_n$) and dense. The main result is that whenever the increasing processes $I_{t\ge T_n}$ have absolutely continuous compensators (in which case the same property holds for any stopping time $T$ whose graph is contained in $B$), then the random set $B$ can be represented as the union of all the points of countably many independent standard Poisson processes (intuitively, a Poisson measure whose rate is $+\infty$ times Lebesgue measure). This may require, however, an innocuous enlargement of filtration. Another characterization of such random sets is roughly that they do not intersect previsible sets of zero Lebesgue measure. Note also an interesting example of a set optional w.r.t. two filtrations, but not w.r.t. their intersection

Keywords: Poisson point processes

Nature: Original

Retrieve article from Numdam

XV: 24, 320-346, LNM 850 (1981)

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

The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in 1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)

Keywords: General filtrations, Strong supermartingales, Snell's envelope, Section theorems

Nature: Original

Retrieve article from Numdam

XV: 25, 347-350, LNM 850 (1981)

Surmartingales-mesures (Martingale theory)

Consider a discrete filtration $({\cal F}_n)$ and let ${\cal A}$ be the algebra, $\cup_n {\cal F}_n$, generating a $\sigma$-algebra ${\cal F}_\infty$. A positive supermartingale $(X_n)$ is called a supermartingale measure if the set function $A\mapsto\lim_n\int_A X_n\,dP$ on $A$ is $\sigma$-additive, and thus can be extended to a measure $\mu$. Then the Lebesgue decomposition of this measure is described (theorem 1). More generally, the Lebesgue decomposition of any measure $\mu$ on ${\cal F}_\infty$ is described. This is meant to complete theorem III.1.5 in Neveu,

Comment: The author points out at the end that theorem 2 had been already proved by Horowitz (

Keywords: Supermartingales, Kunita decomposition

Nature: Original

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XV: 26, 351-370, LNM 850 (1981)

Mesurabilité des débuts et théorème de section~: le lot à la portée de toutes les bourses (General theory of processes)

One of the main topics in these seminars has been the application to stochastic processes of results from descriptive set theory and capacity theory, at different levels. Since these results are considered difficult, many attempts have been made to shorten and simplify the exposition. A noteworthy one was 511, in which Dellacherie introduced ``rabotages'' (306) to develop the theory without analytic sets; see also 1246, 1255. The main feature of this paper is a new interpretation of rabotages as a two-persons game, ascribed to Telgarsky though no reference is given, leading to a pleasant exposition of the whole theory and its main applications

Keywords: Section theorems, Capacities, Sierpinski's ``rabotages''

Nature: Original

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XV: 27, 371-387, LNM 850 (1981)

Sur les noyaux $\sigma$-finis (Measure theory)

This paper is an improvement of 1235. Assume $(X,{\cal X})$ and $(Y,{\cal Y})$ are measurable spaces and $m(x,A)$ is a kernel, i.e., is measurable in $x\in X$ for $A\in{\cal Y}$, and is a $\sigma$-finite measure in $A$ for $x\in X$. Then the problem is to represent the measures $m(x,dy)$ as $g(x,y)\,N(x,dy)$ where $g$ is a jointly measurable function and $N$ is a Markov kernel---possibly enlarging the $\sigma$-field ${\cal X}$ to include analytic sets. The crucial hypothesis (called

Keywords: Kernels, Radon-Nikodym theorem

Nature: Original

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XV: 28, 388-398, LNM 850 (1981)

Sur les travaux de Krylov en théorie de l'intégrale stochastique (Martingale theory)

The well-known work of Malliavin deals with the existence of smooth densities for solutions of stochastic differential equations with smooth coefficients satisfying a hypoellipticity condition. N.V.~Krylov's earlier work (among many papers see

Keywords: Stochastic integrals, Existence of densities

Nature: Exposition

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XV: 29, 399-412, LNM 850 (1981)

Sur la dérivation stochastique au sens de Davis (Stochastic calculus, Brownian motion)

The problem is that of estimating $H_t$ from a knowledge of the process $\int H_sdX_s$, where $X_t$ is a continuous (semi)martingale, as a limit of ratios of the form $\int_t^{t+\alpha} H_sdX_s/(X_{t+\alpha}-X_t)$, or replacing $t+\alpha$ by suitable stopping times

Comment: The problem was suggested and partially solved by Mark H.A. Davis (

Keywords: Stochastic integrals

Nature: Original

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XV: 30, 413-489, LNM 850 (1981)

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,

Keywords: Semimartingales, Formal semimartingales, Stochastic integrals

Nature: Original

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XV: 31, 490-492, LNM 850 (1981)

Sur deux questions posées par Schwartz (Stochastic calculus)

Schwartz studied semimartingales in random open sets, and raised two questions: Given a semimartingale $X$ and a random open set $A$, 1) Assume $X$ is increasing in every subinterval of $A$; then is $X$ equal on $A$ to an increasing adapted process on the whole line? 2) Same statement with ``increasing'' replaced by ``continuous''. Schwartz could prove statement 1) assuming $X$ was continuous. It is proved here that 1) is false if $X$ is only cadlag, and that 2) is false in general, though it is true if $A$ is previsible, or only accessible

Keywords: Random sets, Semimartingales in a random open set

Nature: Original

Retrieve article from Numdam

XV: 32, 493-498, LNM 850 (1981)

Quasi-martingales et variations (Martingale theory)

This paper contains remarks on quasimartingales, the most useful of which being perhaps the fact that, for a right-continuous process, the stochastic variation is the same with respect to the filtrations $({\cal F}_{t})$ and $({\cal F}_{t-})$

Keywords: Quasimartingales

Nature: Original

Retrieve article from Numdam

XV: 33, 499-522, LNM 850 (1981)

Quelques remarques sur la topologie des semimartingales. Applications aux intégrales stochastiques (Stochastic calculus)

This paper contains a number of useful technical results on the topology of semimartingales (see 1324), some of which were previously known with more complicated proofs. In particular, it is shown how to improve the convergence of sequences of semimartingales by a convenient change of probability. The topology of semimartingales is used to handle elegantly the stochastic integration of previsible processes which are not locally bounded (see 1415). Finally, boundedness of a set of semimartingales is shown to be equivalent to the boundedness (in an elementary sense) of a set of increasing processes controlling them in the sense of Métivier-Pellaumail (see 1412, 1413, 1414)

Keywords: Semimartingales, Stochastic integrals, Spaces of semimartingales, Métivier-Pellaumail inequality

Nature: Original

Retrieve article from Numdam

XV: 34, 523-525, LNM 850 (1981)

Sur la caractérisation des semi-martingales (General theory of processes, Stochastic calculus)

This is a sequel to the preceding paper 1533, giving a simple proof that any semimartingale may be brought into any class ${\cal S}^p$ by a convenient change of probability

Keywords: Semimartingales, Spaces of semimartingales

Nature: Original

Retrieve article from Numdam

XV: 35, 526-528, LNM 850 (1981)

Sur certains commutateurs d'une filtration (General theory of processes)

Let $({\cal F}_t)$ be a filtration satisfying the usual conditions and ${\cal G}$ be a $\sigma$-field. Then the conditional expectation $E[.|{\cal G}]$ commutes with $E[.|{\cal F}_T]$ for all stopping times $T$ if and only if for some stopping time $S$ ${\cal G}$ lies between ${\cal F}_{S-}]$ and ${\cal F}_S]$

Keywords: Conditional expectations

Nature: Original

Retrieve article from Numdam

XV: 36, 529-546, LNM 850 (1981)

Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (Measure theory)

For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $

Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see 1228

Keywords: Fuzzy random variables, Convergence in law

Nature: Original

Retrieve article from Numdam

XV: 37, 547-560, LNM 850 (1981)

Convergence en loi de semimartingales et variation quadratique (General theory of processes, Stochastic calculus)

The convergence in law of cadlag processes to a cadlag process being understood in the sense of Skorohod, the problem is to find sufficient conditions under which, given semimartingales $X^n$ and $X$ such that $X^n\rightarrow X$ in law, one may deduce that $[X^n,X^n]$ converges in law to $[X,X]$. This is achieved assuming a uniform bound on the expectations of the supremum of the jumps. A version of the theorem applied to processes which are not semimartingales, but are equal to semimartingales on large sets

Keywords: Semimartingales, Skorohod topology, Convergence in law

Nature: Original

Retrieve article from Numdam

XV: 38, 561-586, LNM 850 (1981)

Solutions faibles et semi-martingales (Stochastic calculus, General theory of processes)

From the author's summary: ``we consider a stochastic differential equation $dX=a(X)\,dZ$ where $Z$ is a semimartingale and $a$ is a previsible functional which is continuous for the uniform norm. We prove the existence of a weak solution for such an equation''. The important point is the definition of a weak solution: it turns out to be a ``fuzzy process'' in the sense of 1536, i.e., a fuzzy r.v. taking values in the Polish space of cadlag sample functions

Keywords: Stochastic differential equations, Weak solutions, Fuzzy random variables

Nature: Original

Retrieve article from Numdam

XV: 39, 587-589, LNM 850 (1981)

Non-confluence des solutions d'une équation stochastique lipschitzienne (Stochastic calculus)

This paper proves that the solutions of a stochastic differential equation $dX_t=f(., t,X_t)\,dM_t$ driven by a continuous semimartingale $M$, where $f(\omega,t,x)$ is as usual previsible in $\omega$ and Lipschitz in $x$, are non-confluent, i.e., the solutions starting at different points never meet

Comment: See also 1506, 1507 (for less general s.d.e.'s), and 1624

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

Nature: Original

Retrieve article from Numdam

XV: 40, 590-603, LNM 850 (1981)

Some remarkable martingales (Martingale theory)

This is a sequel to a well-known paper by the authors (

Keywords: Pure martingales, Previsible representation

Nature: Original

Retrieve article from Numdam

XV: 41, 604-617, LNM 850 (1981)

Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)

This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case

Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps

Nature: Original

Retrieve article from Numdam

XV: 42, 618-626, LNM 850 (1981)

Processus ponctuels marqués stochastiques. Représentation des martingales et filtration naturelle quasicontinue à gauche (General theory of processes)

This paper contains a study of the filtration generated by a point process (multivariate: it takes values in a Polish space), and in particular of its quasi-left continuity, and previsible representation

Keywords: Point processes, Previsible representation

Nature: Original

Retrieve article from Numdam

XV: 43, 627-631, LNM 850 (1981)

Some remarks on processes with independent increments (Independent increments)

This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also 1544

Keywords: Poisson processes, Lévy measures

Nature: Original

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XV: 44, 632-642, LNM 850 (1981)

Mesures à accroissements indépendants et P.A.I. non homogènes (Independent increments)

This is an improved version of 1310: the classical theorem of Lévy on the structure of processes with independent increments is elegantly proved by martingale methods, in the non-homogeneous case, and it is proved that the process is a special semimartingale if and only if it is integrable

Nature: New proof of known results

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XV: 45, 643-668, LNM 850 (1981)

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (General theory of processes, Brownian motion, Martingale theory)

This is a sequel to 1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in 1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given

Comment: Further results are given by Malric

Nature: Original

Retrieve article from Numdam

XV: 46, 669-670, LNM 850 (1981)

Une remarque sur les lois de certains temps d'atteinte (Brownian motion)

Let $T$ be the exit time of the interval $[-d,c]$ for a Brownian motion starting at $0$. A classical formula giving the Laplace transform of the law of $T$ can be extended by analytical continuation to the positive axis. It is shown here that this extension has a purely probabilistic proof. The same method gives two other formulas

Keywords: Exit time from an interval

Nature: New proof of known results

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XV: 47, 671-672, LNM 850 (1981)

Une remarque sur les semi-martingales à deux indices (Several parameter processes)

Let $({\cal F}^1_s)$ and $({\cal F}^2_t)$ be two filtrations whose conditional expectations commute. Let $(A_t)$ be a bounded increasing process adapted to $({\cal F}^2_t)$. It had been proved under stringent absolute continuity conditions on $A$ that the process $X_{st}=E[A_t\,|\,{\cal F}^1_s]$ was a semimartingale (a stochastic integrator). A counterexample is given here to show that this is not true in general

Keywords: Two-parameter semimartingales

Nature: Original

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XV: 48, 673-688, LNM 850 (1981)

Un exemple de processus à deux indices sans l'hypothèse F4 (Several parameter processes)

A natural two-parameter filtration is associated with a random point in the positive quadrant $

Keywords: Two-parameter processes

Nature: Original

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