Quick search | Browse volumes | |

XIII: 01, 1-3, LNM 721 (1979)

**BORELL, Christer**

On the integrability of Banach space valued Walsh polynomials (Banach space valued random variables)

The $L^2$ space over the standard Bernoulli measure on $\{-1,1\}^**N**$ has a well-known orthogonal basis $(e_\alpha)$ indexed by the finite subsets of $**N**$. The Walsh polynomials of order $d$ with values in a Banach space $E$ are linear combinations $\sum_\alpha c_\alpha e_\alpha$ where $c_\alpha\in E$ and $\alpha$ is a finite subset with $d$ elements. It is shown that on this space (as on the Wiener chaos spaces) all $L^p$ norms are equivalent with precise bounds, for $1<p<\infty$. The proof uses the discrete version of hypercontractivity

Keywords: Walsh polynomials, Hypercontractivity

Nature: Original

Retrieve article from Numdam

XIII: 02, 4-21, LNM 721 (1979)

**CHATTERJI, Shrishti Dhav**

Le principe des sous-suites dans les espaces de Banach (Banach space valued random variables)

The ``principle of subsequences'' investigated in the author's paper 604 says roughly that any suitably bounded sequence of r.v.'s contains a subsequence which in some respect ``looks like'' a sequence of i.i.d. random variables. Extensions are considered here in the case of Banach space valued random variables. The paper has the character of a preliminary investigation, though several non-trivial results are indicated (one of them in the Hilbert space case)

Keywords: Subsequences

Nature: Original

Retrieve article from Numdam

XIII: 03, 22-40, LNM 721 (1979)

**GINÉ, Evarist**

Domains of attraction in Banach spaces (Banach space valued random variables)

To be completed

Comment: A correction is given as 1402

Nature: Original

Retrieve article from Numdam

XIII: 04, 41-71, LNM 721 (1979)

**LAPRESTÉ, Jean-Thierry**

Charges, poids et mesures de Lévy dans les espaces vectoriels localement convexes (Banach space valued random variables)

To be completed

Nature: Original

Retrieve article from Numdam

XIII: 05, 72-89, LNM 721 (1979)

**MARCUS, Michael B.**; **PISIER, Gilles**

Random Fourier series on locally compact abelian groups (Banach space valued random variables, Harmonic analysis)

To be completed

Nature: Original

Retrieve article from Numdam

XIII: 06, 90-115, LNM 721 (1979)

**AZÉMA, Jacques**; **YOR, Marc**

Une solution simple au problème de Skorokhod (Brownian motion)

An explicit solution is given to Skorohod's problem: given a distribution $\mu$ with mean $0$ and finite second moment $\sigma^2$, find a (non randomized) stopping time $T$ of a Brownian motion $(X_t)$ such that $X_T$ has the distribution $\mu$ and $E[T]=\sigma^2$. It is shown that if $S_t$ is the one-sided supremum of $X$ at time $t$, $T=\inf\{t:S_t\ge\psi(X_t)\}$ solves the problem, where $\psi(x)$ is the barycenter of $\mu$ restricted to $[x,\infty[$. The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray-Knight theorem on local times

Comment: The subject is further investigated in 1356 and 1441. See also 1515. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIII: 07, 116-117, LNM 721 (1979)

**ÉMERY, Michel**; **STRICKER, Christophe**

Démonstration élémentaire d'un résultat d'Azéma et Jeulin (Martingale theory)

A short proof is given for the following result: given a positive supermartingale $X$ and $h>0$, the supermartingale $E[X_{t+h}\,|\,{\cal F}_t]$ belongs to the class (D). The original proof (*Ann. Inst. Henri Poincaré,* **12**, 1976) used Föllmer's measures

Keywords: Class (D) processes

Nature: Original

Retrieve article from Numdam

XIII: 08, 118-125, LNM 721 (1979)

**YOEURP, Chantha**

Sauts additifs et sauts multiplicatifs des semi-martingales (Martingale theory, General theory of processes)

First of all, the jump processes of special semimartingales are characterized (using a result of 1121, 1129 on the jump processes of local martingales). Then a similar problem is solved for multiplicative jumps, a result which includes that of Garcia and al. 1206. A technical lemma characterizes optional processes, bounded in $L^1$, whose previsible projection vanishes

Keywords: Jump processes

Nature: Original

Retrieve article from Numdam

XIII: 09, 126-131, LNM 721 (1979)

**PRATELLI, Maurizio**

Le support exact du temps local d'une martingale continue (Martingale theory)

It is well known in the Brownian case that the zero set and the support of the local time are the same. For a continuous local martingale $(X_t)$ with zero set $H$ and local time $(L_t)$, it is shown that the support of $dL$ is exactly the perfect kernel of the boundary of $H$

Keywords: Local times

Nature: Original

Retrieve article from Numdam

XIII: 10, 132-137, LNM 721 (1979)

**SIDIBÉ, Ramatoulaye**

Martingales locales à accroissements indépendants (Martingale theory, Independent increments)

It is shown here that a process with (stationary) independent increments which is a local martingale must be a true martingale

Comment: The case of non-stationary increments is considered in 1544. See also the errata sheet of vol. XV

Keywords: Local martingales, Lévy processes

Nature: Original

Retrieve article from Numdam

XIII: 11, 138-141, LNM 721 (1979)

**REBOLLEDO, Rolando**

Décomposition des martingales locales et raréfaction des sauts (Martingale theory)

The general topic underlying this paper is that of convergence in law of a sequence of local martingales $M^n$ to a continuous Gaussian local martingale, i.e., a result analogue to the Central Limit Theorem in the Skorohod topology. This rests on three properties: tightness, convergence of the processes $<M^n,M^n>_t$ to a deterministic process, and a property of ``rarefaction of jumps''. The paper is devoted to a general discussion of the latter property

Comment: A correction is given as 1430

Keywords: Convergence in law, Tightness

Nature: Original

Retrieve article from Numdam

XIII: 12, 142-161, LNM 721 (1979)

**MÉMIN, Jean**; **SHIRYAEV, Albert N.**

Un critère prévisible pour l'uniforme intégrabilité des semimartingales exponentielles (Martingale theory)

A condition is given so that the stochastic exponential of a special semimartingale $X$ is a uniformly integrable process. It involves only the local characteristics of $X$, i.e., its previsible compensator, Lévy measure, and quadratic variation of the continuous martingale part. The proof rests on multiplicative decompositions, and known results in the case of martingales

Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition, Local characteristics

Nature: Original

Retrieve article from Numdam

XIII: 13, 162-173, LNM 721 (1979)

**CAIROLI, Renzo**

Sur la convergence des martingales indexées par ${\bf N}\times{\bf N}$ (Several parameter processes)

For two parameter (discrete) martingales, it is known that uniform integrability does not imply a.s. convergence. But if all (discrete) martingale transforms by indicators of previsible sets are uniformly integrable, then a.s. convergence obtains

Keywords: Almost sure convergence, Martingale transforms

Nature: Original

Retrieve article from Numdam

XIII: 14, 174-198, LNM 721 (1979)

**CAIROLI, Renzo**; **GABRIEL, Jean-Pierre**

Arrêt de certaines suites multiples de variables aléatoires indépendantes (Several parameter processes, Independence)

Let $(X_n)$ be independent, identically distributed random variables. It is known that $X_T/T\in L^1$ for all stopping times $T$ (or the same with $S_n=X_1+...+X_n$ replacing $X_n$) if and only if $X\in L\log L$. The problem is to extend this to several dimensions, $**N**^d$ ($d>1$) replacing $**N**$. Then a stopping time $T$ becomes a stopping point, of which two definitions can be given (the past at time $n$ being defined either as the past rectangle, or the complement of the future rectangle), and $|T|$ being defined as the product of the coordinates). The appropriate space then is $L\log L$ or $L\log^d L$ depending on the kind of stopping times involved. Also the integrability of the supremum of the processes along random increasing paths is considered

Keywords: Stopping points, Random increasing paths

Nature: Original

Retrieve article from Numdam

XIII: 15, 199-203, LNM 721 (1979)

**MEYER, Paul-André**

Une remarque sur le calcul stochastique dépendant d'un paramètre (General theory of processes)

Call a ``process'' a measurable function $X(u,t,\omega)$ where $t$ and $\omega$ are as usual and $u$ is a parameter ranging over some nice measurable space ${\cal U}$. Say that $X$ is evanescent if $X(.\,,\,.\,,\omega)\equiv0$ for a.a. $\omega$. The problem is to define previsible processes, and previsible projections defined up to evanescent sets. This is achieved following Jacod,*Zeit. für W-Theorie,* **31**, 1975. The main feature is the corresponding use of random measures, previsible random measures, and previsible dual projections

Keywords: Processes depending on a parameter, Previsible processes, Previsible projections, Random measures

Nature: Exposition, Original additions

Retrieve article from Numdam

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

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

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

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

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

Nature: Original

Retrieve article from Numdam

XIII: 17, 216-226, LNM 721 (1979)

**LETTA, Giorgio**

Quasimartingales et formes linéaires associées (General theory of processes, Martingale theory)

This paper gives a definition of a quasimartingale $X$ different from the usual one using the stochastic variation: the linear functional $E[\int_0^{\infty} H_s dX_s]$ should be relatively bounded on the vector lattice (Riesz space) of elementary previsible processes. Many classical theorems have simple proofs or elegant interpretations in this language

Keywords: Quasimartingales, Riesz spaces

Nature: Original

Retrieve article from Numdam

XIII: 18, 227-232, LNM 721 (1979)

**BRUNEAU, Michel**

Sur la $p$-variation d'une surmartingale continue (Martingale theory)

The $p$-variation of a deterministic function being defined in the obvious way as a supremum over all partitions, the sample functions of a continuous martingale (and therefore semimartingale) are known to be of finite $p$-variation for $p>2$ (not for $p=2$ in general: non-anticipating partitions are not sufficient to compute the $p$-variation). If $X$ is a continuous supermartingale, a universal bound is given on the expected $p$-variation of $X$ on the interval $[0,T_\lambda]$, where $T_\lambda=\inf\{t:|X_t-X_0|\ge\lambda\}$. The main tool is Doob's classical upcrossing inequality

Comment: For an extension see 1319. These properties are used in T.~Lyons' pathwise theory of stochastic differential equations; see his long article in*Rev. Math. Iberoamericana* 14, 1998

Keywords: $p$-variation, Upcrossings

Nature: Original

Retrieve article from Numdam

XIII: 19, 233-237, LNM 721 (1979)

**STRICKER, Christophe**

Sur la $p$-variation des surmartingales (Martingale theory)

The method of the preceding paper of Bruneau 1318 is extended to all right-continuous semimartingales

Keywords: $p$-variation, Upcrossings

Nature: Original

Retrieve article from Numdam

XIII: 20, 238-239, LNM 721 (1979)

**STRICKER, Christophe**

Une remarque sur l'exposé précédent (Martingale theory)

A few comments are added to the preceding paper 1319, concerning in particular its relationship with results of Lépingle,*Zeit. für W-Theorie,* **36**, 1976

Keywords: $p$-variation, Upcrossings

Nature: Original

Retrieve article from Numdam

XIII: 21, 240-249, LNM 721 (1979)

**MEYER, Paul-André**

Représentations multiplicatives de sousmartingales, d'après Azéma (Martingale theory)

The problem of the multiplicative decomposition of a positive supermartingale $Y$ is relatively easy, but the similar problem for a positive submartingale (find a previsible decreasing process $(C_t)$ such that $(C_tY_t)$ is a martingale) is plagued by the zeros of $Y$. An important idea of Azéma (*Zeit. für W-Theorie,* **45**, 1978) is the introduction of a *multiplicative system * as a two-parameter process $C_{st}$ taking values in $[0,1]$, defined for $s\le t$, such that $C_{tt}=1$, $C_{st}C_{tu}=C_{su}$ for $s\le t\le u$, decreasing and previsible in $t$ for fixed $s$, such that $E[C_{st}Y_t\,|\,{\cal F}_s]=Y_s$ for $s<t$. Then for fixed $s$ the process $(C_{st}Y_t)$ turns out to be a right-continuous martingale on $[s,\infty[$, and what we have done amounts to pasting together all the multiplicative decompositions on zero-free intervals. Existence (and uniqueness of multiplicative systems are proved, though the uniqueness result is slightly different from Azéma's

Keywords: Multiplicative decomposition

Nature: Exposition, Original additions

Retrieve article from Numdam

XIII: 22, 250-252, LNM 721 (1979)

**CHOU, Ching Sung**

Caractérisation d'une classe de semimartingales (Martingale theory, Stochastic calculus)

The class of semimartingales $X$ such that the stochastic integral $J\,**.**\,X$ is a martingale for some nowhere vanishing previsible process $J$ is a natural class of martingale-like processes. Local martingales are exactly the members of this class which are special semimartingales

Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997)

Keywords: Local martingales, Stochastic integrals

Nature: Original

Retrieve article from Numdam

XIII: 23, 253-259, LNM 721 (1979)

**SPILIOTIS, Jean**

Sur les intégrales stochastiques de L.C. Young (Stochastic calculus)

This is a partial exposition of a theory of stochastic integration due to L.C. Young (*Advances in Prob.* **3**, 1974)

Keywords: Stochastic integrals

Nature: Exposition

Retrieve article from Numdam

XIII: 24, 260-280, LNM 721 (1979)

**ÉMERY, Michel**

Une topologie sur l'espace des semimartingales (General theory of processes, Stochastic calculus)

The stability theory for stochastic differential equations was developed independently by Émery (*Zeit. für W-Theorie,* **41**, 1978) and Protter (same journal, **44**, 1978). However, these results were stated in the language of convergent subsequences instead of true topological results. Here a linear topology (like convergence in probability: metrizable, complete, not locally convex) is defined on the space of semimartingales. Side results concern the Banach spaces $H^p$ and $S^p$ of semimartingales. Several useful continuity properties are proved

Comment: This topology has become a standard tool. For its main application, see the next paper 1325

Keywords: Semimartingales, Spaces of semimartingales

Nature: Original

Retrieve article from Numdam

XIII: 25, 281-293, LNM 721 (1979)

**ÉMERY, Michel**

Équations différentielles stochastiques lipschitziennes~: étude de la stabilité (Stochastic calculus)

This is the main application of the topologies on processes and semimartingales introduced in 1324. Using a very general definition of stochastic differential equations turns out to make the proof much simpler, and the existence and uniqueness of solutions of such equations is proved anew before the stability problem is discussed. Useful inequalities on stochastic integration are proved, and used as technical tools

Comment: For all of this subject, the book of Protter*Stochastic Integration and Differential Equations,* Springer 1989, is a useful reference

Keywords: Stochastic differential equations, Stability

Nature: Original

Retrieve article from Numdam

XIII: 26, 294-306, LNM 721 (1979)

**BONAMI, Aline**; **LÉPINGLE, Dominique**

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $**P**$ two functionals of a $**P**$-martingale---remains true, possibly with new constants, when $**P**$ is replaced by an equivalent law $Z.**P**$. To this order, the ``weight'' $Z$ must satisfy special conditions, among which a probabilistic version of Muckenhoupt's (1972) $(A_p)$ condition and a condition of multiplicative boundedness on the jumps of the martingale $E[Z\,|\,{\cal F}_t]$. This volume contains three papers on weighted norms inequalities, 1326, 1327, 1328, with considerable overlap. Here the main topic is the weighted-norm extension of the Burkholder-Gundy inequalities

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

XIII: 27, 307-312, LNM 721 (1979)

**IZUMISAWA, Masataka**; **SEKIGUCHI, Takesi**

Weighted norm inequalities for martingales (Martingale theory)

See the review of 1326. The topic is the same, though the proof is different

Comment: See the paper by Kazamaki-Izumisawa in*Tôhoku Math. J.* **29**, 1977. For a modern reference see also Kazamaki, *Continuous Exponential Martingales and $\,BMO$,* LNM. 1579, 1994

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

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

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

Inégalités de normes avec poids (Martingale theory)

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

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

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

Retrieve article from Numdam

XIII: 29, 332-359, LNM 721 (1979)

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

Inégalité de Hardy, semimartingales, et faux-amis (Martingale theory, General theory of processes)

The main purpose of this paper is to warn against ``obvious'' statements which are in fact false. Let $({\cal G}_t)$ be an enlargement of $({\cal F}_t)$. Assume that ${\cal F}$ has the previsible representation property with respect to a martingale $X$ which is a ${\cal G}$-semimartingale. Then it does not follow that every ${\cal F}$-martingale $Y$ is a ${\cal G}$-semimartingale. Also, even if $Y$ is a ${\cal G}$-semimartingale, its ${\cal G}$-compensator may have bad absolute continuity properties. The counterexample to the first statement involves a detailed study of the initial enlargement of the filtration of Brownian motion $(B_t)_{t\le 1}$ by the random variable $B_1$, which transforms it into the Brownian bridge, a semimartingale. Then the stochastic integrals with respect to $B$ which are ${\cal G}$-semimartingales are completely described, and this is the place where the classical Hardy inequality appears

Keywords: Hardy's inequality, Previsible representation

Nature: Original

Retrieve article from Numdam

XIII: 30, 360-370, LNM 721 (1979)

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

Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)

The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$

Comment: On the same topic see 1518

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

Nature: Original

Retrieve article from Numdam

XIII: 31, 371-377, LNM 721 (1979)

**DELLACHERIE, Claude**

Inégalités de convexité pour les processus croissants et les sousmartingales (Martingale theory)

Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,*Probabilités et Potentiels B,* Chapter VI

Keywords: Martingale inequalities, Convex functions

Nature: Exposition, Original proofs

Retrieve article from Numdam

XIII: 32, 378-384, LNM 721 (1979)

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

Théorème de séparation dans le problème d'arrêt optimal (General theory of processes)

Let $({\cal G}_t)$ be an enlargement of a filtration $({\cal F}_t)$ with the property that for every $t$, if $X$ is ${\cal G}_t$-measurable, then $E[X\,|\,{\cal F}_t]=E[X\,|\,{\cal F}_\infty]$. Then if $(X_t)$ is a ${\cal F}$-optional process, its Snell envelope is the same in both filtrations. Applications are given to filtering theory

Keywords: Optimal stopping, Snell's envelope, Filtering theory

Nature: Original

Retrieve article from Numdam

XIII: 33, 385-399, LNM 721 (1979)

**LE JAN, Yves**

Martingales et changement de temps (Martingale theory, Markov processes)

The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to 1108 and 1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary

Keywords: Changes of time, Energy, Douglas formula

Nature: Original

Retrieve article from Numdam

XIII: 34, 400-406, LNM 721 (1979)

**YOR, Marc**

Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)

This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$

Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$

Nature: Original

Retrieve article from Numdam

XIII: 35, 407-426, LNM 721 (1979)

**YOR, Marc**

En cherchant une définition naturelle des intégrales stochastiques optionnelles (Stochastic calculus)

While the stochastic integral of a previsible process is a very natural object, the optional (compensated) stochastic integral is somewhat puzzling: it concerns martingales only, and depends on the probability law. This paper sketches a ``pedagogical'' approach, using a version of Fefferman's inequality for thin processes to characterize those thin processes which are jump processes of local martingales. The results of 1121, 1129 are easily recovered. Then an attempt is made to extend the optional integral to semimartingales

Keywords: Optional stochastic integrals, Fefferman inequality

Nature: Original

Retrieve article from Numdam

XIII: 36, 427-440, LNM 721 (1979)

**YOR, Marc**

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (Brownian motion)

The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property

Comment: Additional results on the same subject appear in 1545 and in Malric*Ann. Inst. H. Poincaré * **26** (1990)

Keywords: Stochastic integrals

Nature: Original

Retrieve article from Numdam

XIII: 37, 441-442, LNM 721 (1979)

**CHOU, Ching Sung**

Démonstration simple d'un résultat sur le temps local (Stochastic calculus)

It follows from Ito's formula that the positive parts of those jumps of a semimartingale $X$ that originate below $0$ are summable. A direct proof is given of this fact

Comment: Though the idea is essentially correct, an embarrassing mistake is corrected as 1429

Keywords: Local times, Semimartingales, Jumps

Nature: Original

Retrieve article from Numdam

XIII: 38, 443-452, LNM 721 (1979)

**EL KAROUI, Nicole**

Temps local et balayage des semimartingales (General theory of processes)

This paper is the first one in a series of reports on the balayage of semimartingales, and the following description is common to all of them. \par Let $H$ be a right-closed optional set, and let $g_t=\sup\{s<t, s\in H\}$ and $D_t=\inf\{s>t,s\in H\}$. Put $L=g_{\infty}$. Let also $G$ be the set of all left-endpoints of intervals contiguous to $H$, i.e., of all points $g_t$ for $t\notin H$. For simplicity we assume here that $D_0=0$ and that $H=\{X=0\}$, where $X$ is a semimartingale with decomposition $X=M+V$, though for a few results (including the balayage formula itself) it is sufficient that $X=0$ on $H$. \par One of the starting points of this paper is the*balayage formula * (see Azéma-Yor, introduction to *Temps Locaux *, *Astérisque *, **52-53**): if $Z$ is a locally bounded previsible process, then $$Z_{g_t}X_t=\int_0^t Z_{g_s}dX_s$$ and therefore $Y_t=Z_{g_t}X_t$ is a semimartingale. The main problem of the series of reports is: what can be said if $Z$ is not previsible, but optional, or even progressive?\par This particular paper is devoted to the study of the non-adapted process $$K_t=\sum_{g\in G,g\le t } (M_{D_g}-M_g)$$ which turns out to have finite variation

Comment: This paper is completed by 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 39, 453-471, LNM 721 (1979)

**YOR, Marc**

Sur le balayage des semi-martingales continues (General theory of processes)

For the general notation, see 1338. This paper is independent from the preceding one 1338, and some overlap occurs. The balayage formula is extended to processes $Z$ which are not locally bounded, and the local time of the semimartingale $Y$ is computed. The class of continuous semimartingales $X$ with canonical decomposition $X=M+V$ such that $dV$ is carried by $H=\{X=0\}$ is introduced and studied. It turns out to be an important class, closely related to ``relative martingales'' (Azéma, Meyer and Yor 2623). A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time $L$ from $H$ leads to three $\sigma$-fields, ${\cal F}_L^p$, ${\cal F}_L^o$, ${\cal F}_L^{\pi}$, and it was considered surprising that the last two could be different (see 1240). Here it is shown that if $X$ is a continuous uniformly integrable martingale with $X_0=0$, $E[X_{\infty}|{\cal F}_L^o]=0\neq E[X_{\infty}|{\cal F}_L^{\pi}]$

Comment: See 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 40, 472-477, LNM 721 (1979)

**STRICKER, Christophe**

Semimartingales et valeur absolue (General theory of processes)

For the general notation, see 1338. A result of Yoeurp that absolute values preserves quasimartingales is extended: convex functions satisfying a Lipschitz condition operate on quasimartingales. For $p\ge1$, $X\in H^p$ implies $|X|^p\in H^1$. Then it is shown that for a continuous adapted process $X$, it is equivalent to say that $X$ and $|X|$ are quasimartingales (or semimartingales). Then comes a result related to the main problem of this series: with the general notations above, if $X$ is assumed to be a quasimartingale such that $X_{D_t}=0$ for all $t$, if the process $Z$ is progressive and bounded, then the process $Z_{g_t}X_t$ is a quasimartingale

Comment: A complement is given in the next paper 1341. See also 1351

Keywords: Balayage, Quasimartingales

Nature: Original

Retrieve article from Numdam

XIII: 41, 478-487, LNM 721 (1979)

**MEYER, Paul-André**; **STRICKER, Christophe**; **YOR, Marc**

Sur une formule de la théorie du balayage (General theory of processes)

For the notation, see the review of 1340. It is shown here that under the same hypotheses, the semimartingale $Z_{g_t}X_t$ is a sum of three terms: the stochastic integral $\int_0^t \zeta_s dX_s$, where $\zeta$ is the previsible projection of $Z$, an explicit sum of jumps involving $Z-\zeta$, and a mysterious continuous process with finite variation $(R_t)$ such that $dR_t$ is carried by $H$, equal to $0$ if $Z$ was optional

Comment: See 1351, 1357

Keywords: Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 42, 488-489, LNM 721 (1979)

**MEYER, Paul-André**

Construction de semimartingales s'annulant sur un ensemble donné (General theory of processes)

The title states exactly the subject of this short report, whose conclusion is: in the Brownian filtration, every closed optional set is the set of zeros of a continuous semimartingale

Keywords: Semimartingales

Nature: Original

Retrieve article from Numdam

XIII: 43, 490-494, LNM 721 (1979)

**WILLIAMS, David**

Conditional excursion theory (Brownian motion, Markov processes)

To be completed

Keywords: Excursions

Nature: Original

Retrieve article from Numdam

XIII: 44, 495-520, LNM 721 (1979)

**BISMUT, Jean-Michel**

Problèmes à frontière libre et arbres de mesures (Miscellanea, Markov processes)

An optimization problem is discussed, in which one is free to choose at any time among three different transition semi-groups

Keywords: Control theory

Nature: Original

Retrieve article from Numdam

XIII: 45, 521-532, LNM 721 (1979)

**JEULIN, Thierry**

Un théorème de J.W. Pitman (Brownian motion, Diffusion theory)

This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$

Keywords: Bessel processes

Nature: New proof of known results

Retrieve article from Numdam

XIII: 46, 533-547, LNM 721 (1979)

**NANOPOULOS, Photius**

Mesures de probabilités sur les entiers et ensembles progressions (Miscellanea)

To be completed

Nature: Original

Retrieve article from Numdam

XIII: 47, 548-556, LNM 721 (1979)

**FUJISAKI, Masatoshi**

On the uniqueness of optimal controls (Miscellanea)

``In section 1 we can give simple criteria for the uniqueness of the optimal controls whose existence is proved by Ikeda-Watanabe in the completely observable case,*Osaka Math. J.*, **14**, 1977. In section 2 we consider the same problem in the partially observable case.'' (From the author's summary)

Keywords: Control theory

Nature: Original

Retrieve article from Numdam

XIII: 48, 557-569, LNM 721 (1979)

**CARMONA, René**

Processus de diffusion gouverné par la forme de Dirichlet de l'opérateur de Schrödinger (Diffusion theory)

Standard conditions on the potential $V$ imply that the Schrödinger operator $-(1/2)ėlta+V$ (when suitably interpreted) is essentially self-adjoint on $L^2(**R**^n,dx)$. Assume it has a ground state $\psi$. Then transferring everything on the Hilbert space $L^2(\mu)$ where $\mu$ has the density $\psi^2$ the operator becomes formally $Df=(-1/2)ėlta f + \nabla h.\nabla f$ where $h=-log\psi$. A problem which has aroused some excitement ( due in part to Nelson's ``stochastic mechanics'') was to construct true diffusions governed by this generator, whose meaning is not even clearly defined unless $\psi$ satisfies regularity conditions, unnatural in this problem. Here a reasonable positive answer is given

Comment: This problem, though difficult, is but the simplest case in Nelson's theory. In this seminar, see 1901, 1902, 2019. Seemingly definitive results on this subject are due to E.~Carlen,*Comm. Math. Phys.*, **94**, 1984. A recent reference is Aebi, *Schrödinger Diffusion Processes,* Birkhäuser 1995

Keywords: Nelson's stochastic mechanics, Schrödinger operators

Nature: Original

Retrieve article from Numdam

XIII: 49, 570-573, LNM 721 (1979)

**CARMONA, René**

Opérateur de Schrödinger à résolvante compacte (Miscellanea)

A sufficient condition for a Schrödinger operator $(-1/2)ėlta+V$ to have a compact resolvent is proved, using standard properties of Brownian paths

Keywords: Schrödinger operators

Nature: Original

Retrieve article from Numdam

XIII: 50, 574-609, LNM 721 (1979)

**JEULIN, Thierry**

Grossissement d'une filtration et applications (General theory of processes, Markov processes)

This is a sequel to the papers 1209 and 1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths

Keywords: Enlargement of filtrations, Williams decomposition

Nature: Original

Retrieve article from Numdam

XIII: 51, 610-610, LNM 721 (1979)

**STRICKER, Christophe**

Encore une remarque sur la ``formule de balayage'' (General theory of processes)

A slight extension of 1341

Keywords: Balayage

Nature: Original

Retrieve article from Numdam

XIII: 52, 611-613, LNM 721 (1979)

**MEYER, Paul-André**

Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (Martingale theory)

In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (*Ann. Prob.* **8**, 1980) to develop the whole theory of stochastic differential equations

Keywords: Doob's inequality, Stochastic differential equations

Nature: Exposition

Retrieve article from Numdam

XIII: 53, 614-619, LNM 721 (1979)

**YOEURP, Chantha**

Solution explicite de l'équation $Z_t=1+\int_0^t |Z_{s-}|\,dX_s$ (Stochastic calculus)

The title describes completely the paper

Keywords: Stochastic differential equations

Nature: Original

Retrieve article from Numdam

XIII: 54, 620-623, LNM 721 (1979)

**MEYER, Paul-André**

Caractérisation des semimartingales, d'après Dellacherie (Stochastic calculus)

This short paper contains the proof of a very important theorem, due to Dellacherie (with the crucial help of Mokobodzki for the functional analytic part). Namely, semimartingales are exactly the processes which give rise to a nice vector measure on the previsible $\sigma$-field, with values in the (non locally convex) space $L^0$. It is only fair to say that this direction was initiated by Métivier and Pellaumail, and that the main result was independently discovered by Bichteler,*Ann. Prob.* **9**, 1981)

Comment: An important lemma which simplifies the proof and has other applications is given by Yan in 1425

Keywords: Semimartingales, Stochastic integrals

Nature: Exposition

Retrieve article from Numdam

XIII: 55, 624-624, LNM 721 (1979)

**YOR, Marc**

Un exemple de J. Pitman (General theory of processes)

The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale $X$, namely martingales of the form $Z_{g_t}X_t$ where $Z$ is previsible. Taking $X$ to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form

Comment: The general problem of finding all martingales which vanish on the zeros of a given continuous martingale is discussed by Azéma and Yor in 2622

Keywords: Balayage, Balayage formula

Nature: Exposition

Retrieve article from Numdam

XIII: 56, 625-633, LNM 721 (1979)

**AZÉMA, Jacques**; **YOR, Marc**

Le problème de Skorokhod~: compléments à l'exposé précédent (Brownian motion)

What the title calls ``the preceding talk'' is 1306. The method is extended to (centered) measures possessing a moment of order one instead of two, preserving the uniform integrability of the stopped martingale

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIII: 57, 634-641, LNM 721 (1979)

**EL KAROUI, Nicole**

A propos de la formule d'Azéma-Yor (General theory of processes)

For the problem and notation, see the review of 1340. The problem is completely solved here, the process $Z_{g_t}X_t$ being represented as the sum of $\int_0^t Z_{g_s}dX_s$ interpreted in a generalized sense ($Z$ being progressive!) and a remainder which can be explicitly written (using optional dual projections of non-adapted processes)

Comment: This paper ends happily the whole series of papers on balayage in this volume

Keywords: Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 58, 642-645, LNM 721 (1979)

**MAISONNEUVE, Bernard**

Martingales de valeur absolue donnée, d'après Protter-Sharpe (Martingale theory)

The main difficulty of Gilat's theorem (every positive submartingale $X$ can be interpreted as the absolute value of a martingale, in a suitably enlarged filtration) is due to the zeros of $X$. In the strictly positive case a simple proof was given by Protter and Sharpe (*Ann. Prob.*, **7**, 1979). This proof is further simplified and slightly generalized

Comment: See also 1407

Keywords: Gilat's theorem

Nature: Exposition, Original additions

Retrieve article from Numdam

XIII: 59, 646-646, LNM 721 (1979)

**BARLOW, Martin T.**

On the left endpoints of Brownian excursions (Brownian motion, Excursion theory)

It is shown that no expansion of the Brownian filtration can be found such that $B_t$ remains a semimartingale, and the set of left endpoints of Brownian excursions becomes optional

Keywords: Progressive sets

Nature: Original

Retrieve article from Numdam

XIII: 60, 647-647, LNM 721 (1979)

**BRETAGNOLLE, Jean**; **HUBER, Catherine**

Corrections à un exposé antérieur (Mathematical statistics)

Two misprints and a more substantial error (in the proof of proposition 1) of 1224 are corrected

Comment: A revised version appeared in (*Zeit. für W-Theorie,* **47**, 1979)

Keywords: Empirical distribution function, Prohorov distance

Nature: Correction

Retrieve article from Numdam

On the integrability of Banach space valued Walsh polynomials (Banach space valued random variables)

The $L^2$ space over the standard Bernoulli measure on $\{-1,1\}^

Keywords: Walsh polynomials, Hypercontractivity

Nature: Original

Retrieve article from Numdam

XIII: 02, 4-21, LNM 721 (1979)

Le principe des sous-suites dans les espaces de Banach (Banach space valued random variables)

The ``principle of subsequences'' investigated in the author's paper 604 says roughly that any suitably bounded sequence of r.v.'s contains a subsequence which in some respect ``looks like'' a sequence of i.i.d. random variables. Extensions are considered here in the case of Banach space valued random variables. The paper has the character of a preliminary investigation, though several non-trivial results are indicated (one of them in the Hilbert space case)

Keywords: Subsequences

Nature: Original

Retrieve article from Numdam

XIII: 03, 22-40, LNM 721 (1979)

Domains of attraction in Banach spaces (Banach space valued random variables)

To be completed

Comment: A correction is given as 1402

Nature: Original

Retrieve article from Numdam

XIII: 04, 41-71, LNM 721 (1979)

Charges, poids et mesures de Lévy dans les espaces vectoriels localement convexes (Banach space valued random variables)

To be completed

Nature: Original

Retrieve article from Numdam

XIII: 05, 72-89, LNM 721 (1979)

Random Fourier series on locally compact abelian groups (Banach space valued random variables, Harmonic analysis)

To be completed

Nature: Original

Retrieve article from Numdam

XIII: 06, 90-115, LNM 721 (1979)

Une solution simple au problème de Skorokhod (Brownian motion)

An explicit solution is given to Skorohod's problem: given a distribution $\mu$ with mean $0$ and finite second moment $\sigma^2$, find a (non randomized) stopping time $T$ of a Brownian motion $(X_t)$ such that $X_T$ has the distribution $\mu$ and $E[T]=\sigma^2$. It is shown that if $S_t$ is the one-sided supremum of $X$ at time $t$, $T=\inf\{t:S_t\ge\psi(X_t)\}$ solves the problem, where $\psi(x)$ is the barycenter of $\mu$ restricted to $[x,\infty[$. The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray-Knight theorem on local times

Comment: The subject is further investigated in 1356 and 1441. See also 1515. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIII: 07, 116-117, LNM 721 (1979)

Démonstration élémentaire d'un résultat d'Azéma et Jeulin (Martingale theory)

A short proof is given for the following result: given a positive supermartingale $X$ and $h>0$, the supermartingale $E[X_{t+h}\,|\,{\cal F}_t]$ belongs to the class (D). The original proof (

Keywords: Class (D) processes

Nature: Original

Retrieve article from Numdam

XIII: 08, 118-125, LNM 721 (1979)

Sauts additifs et sauts multiplicatifs des semi-martingales (Martingale theory, General theory of processes)

First of all, the jump processes of special semimartingales are characterized (using a result of 1121, 1129 on the jump processes of local martingales). Then a similar problem is solved for multiplicative jumps, a result which includes that of Garcia and al. 1206. A technical lemma characterizes optional processes, bounded in $L^1$, whose previsible projection vanishes

Keywords: Jump processes

Nature: Original

Retrieve article from Numdam

XIII: 09, 126-131, LNM 721 (1979)

Le support exact du temps local d'une martingale continue (Martingale theory)

It is well known in the Brownian case that the zero set and the support of the local time are the same. For a continuous local martingale $(X_t)$ with zero set $H$ and local time $(L_t)$, it is shown that the support of $dL$ is exactly the perfect kernel of the boundary of $H$

Keywords: Local times

Nature: Original

Retrieve article from Numdam

XIII: 10, 132-137, LNM 721 (1979)

Martingales locales à accroissements indépendants (Martingale theory, Independent increments)

It is shown here that a process with (stationary) independent increments which is a local martingale must be a true martingale

Comment: The case of non-stationary increments is considered in 1544. See also the errata sheet of vol. XV

Keywords: Local martingales, Lévy processes

Nature: Original

Retrieve article from Numdam

XIII: 11, 138-141, LNM 721 (1979)

Décomposition des martingales locales et raréfaction des sauts (Martingale theory)

The general topic underlying this paper is that of convergence in law of a sequence of local martingales $M^n$ to a continuous Gaussian local martingale, i.e., a result analogue to the Central Limit Theorem in the Skorohod topology. This rests on three properties: tightness, convergence of the processes $<M^n,M^n>_t$ to a deterministic process, and a property of ``rarefaction of jumps''. The paper is devoted to a general discussion of the latter property

Comment: A correction is given as 1430

Keywords: Convergence in law, Tightness

Nature: Original

Retrieve article from Numdam

XIII: 12, 142-161, LNM 721 (1979)

Un critère prévisible pour l'uniforme intégrabilité des semimartingales exponentielles (Martingale theory)

A condition is given so that the stochastic exponential of a special semimartingale $X$ is a uniformly integrable process. It involves only the local characteristics of $X$, i.e., its previsible compensator, Lévy measure, and quadratic variation of the continuous martingale part. The proof rests on multiplicative decompositions, and known results in the case of martingales

Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition, Local characteristics

Nature: Original

Retrieve article from Numdam

XIII: 13, 162-173, LNM 721 (1979)

Sur la convergence des martingales indexées par ${\bf N}\times{\bf N}$ (Several parameter processes)

For two parameter (discrete) martingales, it is known that uniform integrability does not imply a.s. convergence. But if all (discrete) martingale transforms by indicators of previsible sets are uniformly integrable, then a.s. convergence obtains

Keywords: Almost sure convergence, Martingale transforms

Nature: Original

Retrieve article from Numdam

XIII: 14, 174-198, LNM 721 (1979)

Arrêt de certaines suites multiples de variables aléatoires indépendantes (Several parameter processes, Independence)

Let $(X_n)$ be independent, identically distributed random variables. It is known that $X_T/T\in L^1$ for all stopping times $T$ (or the same with $S_n=X_1+...+X_n$ replacing $X_n$) if and only if $X\in L\log L$. The problem is to extend this to several dimensions, $

Keywords: Stopping points, Random increasing paths

Nature: Original

Retrieve article from Numdam

XIII: 15, 199-203, LNM 721 (1979)

Une remarque sur le calcul stochastique dépendant d'un paramètre (General theory of processes)

Call a ``process'' a measurable function $X(u,t,\omega)$ where $t$ and $\omega$ are as usual and $u$ is a parameter ranging over some nice measurable space ${\cal U}$. Say that $X$ is evanescent if $X(.\,,\,.\,,\omega)\equiv0$ for a.a. $\omega$. The problem is to define previsible processes, and previsible projections defined up to evanescent sets. This is achieved following Jacod,

Keywords: Processes depending on a parameter, Previsible processes, Previsible projections, Random measures

Nature: Exposition, Original additions

Retrieve article from Numdam

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

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

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

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

Nature: Original

Retrieve article from Numdam

XIII: 17, 216-226, LNM 721 (1979)

Quasimartingales et formes linéaires associées (General theory of processes, Martingale theory)

This paper gives a definition of a quasimartingale $X$ different from the usual one using the stochastic variation: the linear functional $E[\int_0^{\infty} H_s dX_s]$ should be relatively bounded on the vector lattice (Riesz space) of elementary previsible processes. Many classical theorems have simple proofs or elegant interpretations in this language

Keywords: Quasimartingales, Riesz spaces

Nature: Original

Retrieve article from Numdam

XIII: 18, 227-232, LNM 721 (1979)

Sur la $p$-variation d'une surmartingale continue (Martingale theory)

The $p$-variation of a deterministic function being defined in the obvious way as a supremum over all partitions, the sample functions of a continuous martingale (and therefore semimartingale) are known to be of finite $p$-variation for $p>2$ (not for $p=2$ in general: non-anticipating partitions are not sufficient to compute the $p$-variation). If $X$ is a continuous supermartingale, a universal bound is given on the expected $p$-variation of $X$ on the interval $[0,T_\lambda]$, where $T_\lambda=\inf\{t:|X_t-X_0|\ge\lambda\}$. The main tool is Doob's classical upcrossing inequality

Comment: For an extension see 1319. These properties are used in T.~Lyons' pathwise theory of stochastic differential equations; see his long article in

Keywords: $p$-variation, Upcrossings

Nature: Original

Retrieve article from Numdam

XIII: 19, 233-237, LNM 721 (1979)

Sur la $p$-variation des surmartingales (Martingale theory)

The method of the preceding paper of Bruneau 1318 is extended to all right-continuous semimartingales

Keywords: $p$-variation, Upcrossings

Nature: Original

Retrieve article from Numdam

XIII: 20, 238-239, LNM 721 (1979)

Une remarque sur l'exposé précédent (Martingale theory)

A few comments are added to the preceding paper 1319, concerning in particular its relationship with results of Lépingle,

Keywords: $p$-variation, Upcrossings

Nature: Original

Retrieve article from Numdam

XIII: 21, 240-249, LNM 721 (1979)

Représentations multiplicatives de sousmartingales, d'après Azéma (Martingale theory)

The problem of the multiplicative decomposition of a positive supermartingale $Y$ is relatively easy, but the similar problem for a positive submartingale (find a previsible decreasing process $(C_t)$ such that $(C_tY_t)$ is a martingale) is plagued by the zeros of $Y$. An important idea of Azéma (

Keywords: Multiplicative decomposition

Nature: Exposition, Original additions

Retrieve article from Numdam

XIII: 22, 250-252, LNM 721 (1979)

Caractérisation d'une classe de semimartingales (Martingale theory, Stochastic calculus)

The class of semimartingales $X$ such that the stochastic integral $J\,

Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997)

Keywords: Local martingales, Stochastic integrals

Nature: Original

Retrieve article from Numdam

XIII: 23, 253-259, LNM 721 (1979)

Sur les intégrales stochastiques de L.C. Young (Stochastic calculus)

This is a partial exposition of a theory of stochastic integration due to L.C. Young (

Keywords: Stochastic integrals

Nature: Exposition

Retrieve article from Numdam

XIII: 24, 260-280, LNM 721 (1979)

Une topologie sur l'espace des semimartingales (General theory of processes, Stochastic calculus)

The stability theory for stochastic differential equations was developed independently by Émery (

Comment: This topology has become a standard tool. For its main application, see the next paper 1325

Keywords: Semimartingales, Spaces of semimartingales

Nature: Original

Retrieve article from Numdam

XIII: 25, 281-293, LNM 721 (1979)

Équations différentielles stochastiques lipschitziennes~: étude de la stabilité (Stochastic calculus)

This is the main application of the topologies on processes and semimartingales introduced in 1324. Using a very general definition of stochastic differential equations turns out to make the proof much simpler, and the existence and uniqueness of solutions of such equations is proved anew before the stability problem is discussed. Useful inequalities on stochastic integration are proved, and used as technical tools

Comment: For all of this subject, the book of Protter

Keywords: Stochastic differential equations, Stability

Nature: Original

Retrieve article from Numdam

XIII: 26, 294-306, LNM 721 (1979)

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

XIII: 27, 307-312, LNM 721 (1979)

Weighted norm inequalities for martingales (Martingale theory)

See the review of 1326. The topic is the same, though the proof is different

Comment: See the paper by Kazamaki-Izumisawa in

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

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

Inégalités de normes avec poids (Martingale theory)

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

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

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

Retrieve article from Numdam

XIII: 29, 332-359, LNM 721 (1979)

Inégalité de Hardy, semimartingales, et faux-amis (Martingale theory, General theory of processes)

The main purpose of this paper is to warn against ``obvious'' statements which are in fact false. Let $({\cal G}_t)$ be an enlargement of $({\cal F}_t)$. Assume that ${\cal F}$ has the previsible representation property with respect to a martingale $X$ which is a ${\cal G}$-semimartingale. Then it does not follow that every ${\cal F}$-martingale $Y$ is a ${\cal G}$-semimartingale. Also, even if $Y$ is a ${\cal G}$-semimartingale, its ${\cal G}$-compensator may have bad absolute continuity properties. The counterexample to the first statement involves a detailed study of the initial enlargement of the filtration of Brownian motion $(B_t)_{t\le 1}$ by the random variable $B_1$, which transforms it into the Brownian bridge, a semimartingale. Then the stochastic integrals with respect to $B$ which are ${\cal G}$-semimartingales are completely described, and this is the place where the classical Hardy inequality appears

Keywords: Hardy's inequality, Previsible representation

Nature: Original

Retrieve article from Numdam

XIII: 30, 360-370, LNM 721 (1979)

Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)

The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$

Comment: On the same topic see 1518

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

Nature: Original

Retrieve article from Numdam

XIII: 31, 371-377, LNM 721 (1979)

Inégalités de convexité pour les processus croissants et les sousmartingales (Martingale theory)

Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,

Keywords: Martingale inequalities, Convex functions

Nature: Exposition, Original proofs

Retrieve article from Numdam

XIII: 32, 378-384, LNM 721 (1979)

Théorème de séparation dans le problème d'arrêt optimal (General theory of processes)

Let $({\cal G}_t)$ be an enlargement of a filtration $({\cal F}_t)$ with the property that for every $t$, if $X$ is ${\cal G}_t$-measurable, then $E[X\,|\,{\cal F}_t]=E[X\,|\,{\cal F}_\infty]$. Then if $(X_t)$ is a ${\cal F}$-optional process, its Snell envelope is the same in both filtrations. Applications are given to filtering theory

Keywords: Optimal stopping, Snell's envelope, Filtering theory

Nature: Original

Retrieve article from Numdam

XIII: 33, 385-399, LNM 721 (1979)

Martingales et changement de temps (Martingale theory, Markov processes)

The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to 1108 and 1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary

Keywords: Changes of time, Energy, Douglas formula

Nature: Original

Retrieve article from Numdam

XIII: 34, 400-406, LNM 721 (1979)

Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)

This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$

Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$

Nature: Original

Retrieve article from Numdam

XIII: 35, 407-426, LNM 721 (1979)

En cherchant une définition naturelle des intégrales stochastiques optionnelles (Stochastic calculus)

While the stochastic integral of a previsible process is a very natural object, the optional (compensated) stochastic integral is somewhat puzzling: it concerns martingales only, and depends on the probability law. This paper sketches a ``pedagogical'' approach, using a version of Fefferman's inequality for thin processes to characterize those thin processes which are jump processes of local martingales. The results of 1121, 1129 are easily recovered. Then an attempt is made to extend the optional integral to semimartingales

Keywords: Optional stochastic integrals, Fefferman inequality

Nature: Original

Retrieve article from Numdam

XIII: 36, 427-440, LNM 721 (1979)

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (Brownian motion)

The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property

Comment: Additional results on the same subject appear in 1545 and in Malric

Keywords: Stochastic integrals

Nature: Original

Retrieve article from Numdam

XIII: 37, 441-442, LNM 721 (1979)

Démonstration simple d'un résultat sur le temps local (Stochastic calculus)

It follows from Ito's formula that the positive parts of those jumps of a semimartingale $X$ that originate below $0$ are summable. A direct proof is given of this fact

Comment: Though the idea is essentially correct, an embarrassing mistake is corrected as 1429

Keywords: Local times, Semimartingales, Jumps

Nature: Original

Retrieve article from Numdam

XIII: 38, 443-452, LNM 721 (1979)

Temps local et balayage des semimartingales (General theory of processes)

This paper is the first one in a series of reports on the balayage of semimartingales, and the following description is common to all of them. \par Let $H$ be a right-closed optional set, and let $g_t=\sup\{s<t, s\in H\}$ and $D_t=\inf\{s>t,s\in H\}$. Put $L=g_{\infty}$. Let also $G$ be the set of all left-endpoints of intervals contiguous to $H$, i.e., of all points $g_t$ for $t\notin H$. For simplicity we assume here that $D_0=0$ and that $H=\{X=0\}$, where $X$ is a semimartingale with decomposition $X=M+V$, though for a few results (including the balayage formula itself) it is sufficient that $X=0$ on $H$. \par One of the starting points of this paper is the

Comment: This paper is completed by 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 39, 453-471, LNM 721 (1979)

Sur le balayage des semi-martingales continues (General theory of processes)

For the general notation, see 1338. This paper is independent from the preceding one 1338, and some overlap occurs. The balayage formula is extended to processes $Z$ which are not locally bounded, and the local time of the semimartingale $Y$ is computed. The class of continuous semimartingales $X$ with canonical decomposition $X=M+V$ such that $dV$ is carried by $H=\{X=0\}$ is introduced and studied. It turns out to be an important class, closely related to ``relative martingales'' (Azéma, Meyer and Yor 2623). A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time $L$ from $H$ leads to three $\sigma$-fields, ${\cal F}_L^p$, ${\cal F}_L^o$, ${\cal F}_L^{\pi}$, and it was considered surprising that the last two could be different (see 1240). Here it is shown that if $X$ is a continuous uniformly integrable martingale with $X_0=0$, $E[X_{\infty}|{\cal F}_L^o]=0\neq E[X_{\infty}|{\cal F}_L^{\pi}]$

Comment: See 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 40, 472-477, LNM 721 (1979)

Semimartingales et valeur absolue (General theory of processes)

For the general notation, see 1338. A result of Yoeurp that absolute values preserves quasimartingales is extended: convex functions satisfying a Lipschitz condition operate on quasimartingales. For $p\ge1$, $X\in H^p$ implies $|X|^p\in H^1$. Then it is shown that for a continuous adapted process $X$, it is equivalent to say that $X$ and $|X|$ are quasimartingales (or semimartingales). Then comes a result related to the main problem of this series: with the general notations above, if $X$ is assumed to be a quasimartingale such that $X_{D_t}=0$ for all $t$, if the process $Z$ is progressive and bounded, then the process $Z_{g_t}X_t$ is a quasimartingale

Comment: A complement is given in the next paper 1341. See also 1351

Keywords: Balayage, Quasimartingales

Nature: Original

Retrieve article from Numdam

XIII: 41, 478-487, LNM 721 (1979)

Sur une formule de la théorie du balayage (General theory of processes)

For the notation, see the review of 1340. It is shown here that under the same hypotheses, the semimartingale $Z_{g_t}X_t$ is a sum of three terms: the stochastic integral $\int_0^t \zeta_s dX_s$, where $\zeta$ is the previsible projection of $Z$, an explicit sum of jumps involving $Z-\zeta$, and a mysterious continuous process with finite variation $(R_t)$ such that $dR_t$ is carried by $H$, equal to $0$ if $Z$ was optional

Comment: See 1351, 1357

Keywords: Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 42, 488-489, LNM 721 (1979)

Construction de semimartingales s'annulant sur un ensemble donné (General theory of processes)

The title states exactly the subject of this short report, whose conclusion is: in the Brownian filtration, every closed optional set is the set of zeros of a continuous semimartingale

Keywords: Semimartingales

Nature: Original

Retrieve article from Numdam

XIII: 43, 490-494, LNM 721 (1979)

Conditional excursion theory (Brownian motion, Markov processes)

To be completed

Keywords: Excursions

Nature: Original

Retrieve article from Numdam

XIII: 44, 495-520, LNM 721 (1979)

Problèmes à frontière libre et arbres de mesures (Miscellanea, Markov processes)

An optimization problem is discussed, in which one is free to choose at any time among three different transition semi-groups

Keywords: Control theory

Nature: Original

Retrieve article from Numdam

XIII: 45, 521-532, LNM 721 (1979)

Un théorème de J.W. Pitman (Brownian motion, Diffusion theory)

This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$

Keywords: Bessel processes

Nature: New proof of known results

Retrieve article from Numdam

XIII: 46, 533-547, LNM 721 (1979)

Mesures de probabilités sur les entiers et ensembles progressions (Miscellanea)

To be completed

Nature: Original

Retrieve article from Numdam

XIII: 47, 548-556, LNM 721 (1979)

On the uniqueness of optimal controls (Miscellanea)

``In section 1 we can give simple criteria for the uniqueness of the optimal controls whose existence is proved by Ikeda-Watanabe in the completely observable case,

Keywords: Control theory

Nature: Original

Retrieve article from Numdam

XIII: 48, 557-569, LNM 721 (1979)

Processus de diffusion gouverné par la forme de Dirichlet de l'opérateur de Schrödinger (Diffusion theory)

Standard conditions on the potential $V$ imply that the Schrödinger operator $-(1/2)ėlta+V$ (when suitably interpreted) is essentially self-adjoint on $L^2(

Comment: This problem, though difficult, is but the simplest case in Nelson's theory. In this seminar, see 1901, 1902, 2019. Seemingly definitive results on this subject are due to E.~Carlen,

Keywords: Nelson's stochastic mechanics, Schrödinger operators

Nature: Original

Retrieve article from Numdam

XIII: 49, 570-573, LNM 721 (1979)

Opérateur de Schrödinger à résolvante compacte (Miscellanea)

A sufficient condition for a Schrödinger operator $(-1/2)ėlta+V$ to have a compact resolvent is proved, using standard properties of Brownian paths

Keywords: Schrödinger operators

Nature: Original

Retrieve article from Numdam

XIII: 50, 574-609, LNM 721 (1979)

Grossissement d'une filtration et applications (General theory of processes, Markov processes)

This is a sequel to the papers 1209 and 1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths

Keywords: Enlargement of filtrations, Williams decomposition

Nature: Original

Retrieve article from Numdam

XIII: 51, 610-610, LNM 721 (1979)

Encore une remarque sur la ``formule de balayage'' (General theory of processes)

A slight extension of 1341

Keywords: Balayage

Nature: Original

Retrieve article from Numdam

XIII: 52, 611-613, LNM 721 (1979)

Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (Martingale theory)

In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (

Keywords: Doob's inequality, Stochastic differential equations

Nature: Exposition

Retrieve article from Numdam

XIII: 53, 614-619, LNM 721 (1979)

Solution explicite de l'équation $Z_t=1+\int_0^t |Z_{s-}|\,dX_s$ (Stochastic calculus)

The title describes completely the paper

Keywords: Stochastic differential equations

Nature: Original

Retrieve article from Numdam

XIII: 54, 620-623, LNM 721 (1979)

Caractérisation des semimartingales, d'après Dellacherie (Stochastic calculus)

This short paper contains the proof of a very important theorem, due to Dellacherie (with the crucial help of Mokobodzki for the functional analytic part). Namely, semimartingales are exactly the processes which give rise to a nice vector measure on the previsible $\sigma$-field, with values in the (non locally convex) space $L^0$. It is only fair to say that this direction was initiated by Métivier and Pellaumail, and that the main result was independently discovered by Bichteler,

Comment: An important lemma which simplifies the proof and has other applications is given by Yan in 1425

Keywords: Semimartingales, Stochastic integrals

Nature: Exposition

Retrieve article from Numdam

XIII: 55, 624-624, LNM 721 (1979)

Un exemple de J. Pitman (General theory of processes)

The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale $X$, namely martingales of the form $Z_{g_t}X_t$ where $Z$ is previsible. Taking $X$ to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form

Comment: The general problem of finding all martingales which vanish on the zeros of a given continuous martingale is discussed by Azéma and Yor in 2622

Keywords: Balayage, Balayage formula

Nature: Exposition

Retrieve article from Numdam

XIII: 56, 625-633, LNM 721 (1979)

Le problème de Skorokhod~: compléments à l'exposé précédent (Brownian motion)

What the title calls ``the preceding talk'' is 1306. The method is extended to (centered) measures possessing a moment of order one instead of two, preserving the uniform integrability of the stopped martingale

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIII: 57, 634-641, LNM 721 (1979)

A propos de la formule d'Azéma-Yor (General theory of processes)

For the problem and notation, see the review of 1340. The problem is completely solved here, the process $Z_{g_t}X_t$ being represented as the sum of $\int_0^t Z_{g_s}dX_s$ interpreted in a generalized sense ($Z$ being progressive!) and a remainder which can be explicitly written (using optional dual projections of non-adapted processes)

Comment: This paper ends happily the whole series of papers on balayage in this volume

Keywords: Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 58, 642-645, LNM 721 (1979)

Martingales de valeur absolue donnée, d'après Protter-Sharpe (Martingale theory)

The main difficulty of Gilat's theorem (every positive submartingale $X$ can be interpreted as the absolute value of a martingale, in a suitably enlarged filtration) is due to the zeros of $X$. In the strictly positive case a simple proof was given by Protter and Sharpe (

Comment: See also 1407

Keywords: Gilat's theorem

Nature: Exposition, Original additions

Retrieve article from Numdam

XIII: 59, 646-646, LNM 721 (1979)

On the left endpoints of Brownian excursions (Brownian motion, Excursion theory)

It is shown that no expansion of the Brownian filtration can be found such that $B_t$ remains a semimartingale, and the set of left endpoints of Brownian excursions becomes optional

Keywords: Progressive sets

Nature: Original

Retrieve article from Numdam

XIII: 60, 647-647, LNM 721 (1979)

Corrections à un exposé antérieur (Mathematical statistics)

Two misprints and a more substantial error (in the proof of proposition 1) of 1224 are corrected

Comment: A revised version appeared in (

Keywords: Empirical distribution function, Prohorov distance

Nature: Correction

Retrieve article from Numdam