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X: 08, 104-117, LNM 511 (1976)

**MEYER, Paul-André**; **YOR, Marc**

Sur la théorie de la prédiction, et le problème de décomposition des tribus ${\cal F}^{\circ}_{t+}$ (General theory of processes)

This paper contains another version of Knight's theory (preceding paper 1007) for cadlag process instead of measurable processes. These results then are applied to the pathology of germ fields: a natural measurability conjecture does not hold, and an example is given of a process $X_t$ such that its natural $\sigma$-field ${\cal F}_{1+}$ is not generated by ${\cal F}_{1}$ and the germ-field at $0$ of the process $(X_{1+s})$

Comment: On the pathology of germ fields, see H. von Weizsäcker,*Ann. Inst. Henri Poincaré,* **19**, 1983

Keywords: Prediction theory, Germ fields

Nature: Original

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X: 22, 481-500, LNM 511 (1976)

**YOR, Marc**

Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (Martingale theory, Stochastic calculus)

This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of ``exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up

Keywords: Optional stochastic integrals, Stochastic exponentials, Lévy systems

Nature: Original

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XI: 14, 257-297, LNM 581 (1977)

**YOR, Marc**

Sur les théories du filtrage et de la prédiction (General theory of processes, Markov processes)

To be completed

Comment: MR 57, 10801

Keywords: Filtering theory, Prediction theory

Nature: Original

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XI: 34, 493-501, LNM 581 (1977)

**YOR, Marc**

A propos d'un lemme de Ch. Yoeurp (General theory of processes, Martingale theory)

Yoeurp's lemma is the following: if $A$ is a previsible process of bounded variation, its square bracket $[A,L]$ with any local martingale $L$ is a local martingale. This useful result was not easily accessible, thus a complete proof is given, with several new applications---in particular, this characterizes previsible processes of bounded variation among semimartingales

Keywords: Yoeurp's lemma, Square bracket

Nature: Original

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XI: 35, 502-517, LNM 581 (1977)

**YOR, Marc**

Remarques sur la représentation des martingales comme intégrales stochastiques (Martingale theory)

The main result on the relation between the previsible representation property of a set of local martingales and the extremality of their joint law appeared in a celebrated paper of Jacod-Yor,*Z. für W-theorie,* **38**, 1977. Several concrete applications are given here, in particular a complete proof of a ``folklore'' result on the representation of local martingales of a Lévy process, and a discussion of the commutation problem of 1123

Comment: This is an intermediate paper between the Jacod-Yor results and the definitive version of previsible representation, using the theorem of Douglas, for which see 1221

Keywords: Previsible representation, Extreme points, Independent increments, Lévy processes

Nature: Original

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XI: 36, 518-528, LNM 581 (1977)

**YOR, Marc**

Sur quelques approximations d'intégrales stochastiques (Martingale theory)

The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form

Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe*Publ. RIMS, Kyoto Univ.* **15**, 1979; Bismut, Mécanique Aléatoire, Springer LNM~866, 1981; Meyer 1505

Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals

Nature: Original

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XII: 09, 61-69, LNM 649 (1978)

**YOR, Marc**

Grossissement d'une filtration et semi-martingales~: théorèmes généraux (General theory of processes)

Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called*progressively enlarged * filtration is the smallest one $({\cal G}_t)$ containing $({\cal F}_t)$, and for which $L$ is a stopping time. The enlargement problem consists in describing the semimartingales $X$ of ${\cal F}$ which remain semimartingales in ${\cal G}$, and in computing their semimartingale characteristics. In this paper, it is proved that $X_tI_{\{t< L\}}$ is a semimartingale in full generality, and that $X_tI_{\{t\ge L\}}$ is a semimartingale whenever $L$ is *honest * for $\cal F$, i.e., is the end of an $\cal F$-optional set

Comment: This result was independently discovered by Barlow,*Zeit. für W-theorie,* 44, 1978, which also has a huge intersection with 1211. Complements are given in 1210, and an explicit decomposition formula for semimartingales in 1211

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 11, 78-97, LNM 649 (1978)

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

Grossissement d'une filtration et semi-martingales~: Formules explicites (General theory of processes)

This contains very substantial improvements on 1209, namely, the explicit computation of the characteristics of the semimartingales involved

Comment: For additional results on enlargements, see the two Lecture Notes volumes**833** (T. Jeulin) and **1118**. See also 1350

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 12, 98-113, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**; **YOR, Marc**

Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)

The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)

Keywords: Hardy spaces, $BMO$

Nature: Original

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XII: 21, 265-309, LNM 649 (1978)

**YOR, Marc**; **SAM LAZARO, José de**

Sous-espaces denses dans $L^1$ ou $H^1$ et représentation des martingales (Martingale theory)

This paper was a considerable step in the study of the general martingale problem, i.e., of the set ${\cal P}$ of all laws on a filtered measurable space under which a given set ${\cal N}$ of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R.G. Douglas (*Michigan Math. J.* 11, 1964), and the main technical difference with preceding papers is the systematic use of stochastic integration in $H^1$. The main result can be stated as follows: given a law $P\in{\cal P}$, the set ${\cal N}$ has the previsible representation property, i.e., ${\cal F}_0$ is trivial and stochastic integrals with respect to elements of ${\cal N}$ are dense in $H^1$, if and only if $P$ is an extreme point of ${\cal P}$. Many examples and applications are given

Comment: The second named author's contribution concerns only the appendix on homogeneous martingales

Keywords: Previsible representation, Douglas theorem, Extremal laws

Nature: Original

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XII: 35, 482-488, LNM 649 (1978)

**YOR, Marc**; **MEYER, Paul-André**

Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (Measure theory)

Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability

Comment: The subject is discussed further in 1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the*weak * topology of $L[\infty$, or with the topology of $L^0$

Keywords: Kernels, Radon-Nikodym theorem

Nature: Original

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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XIV: 06, 53-61, LNM 784 (1980)

**AZÉMA, Jacques**; **GUNDY, Richard F.**; **YOR, Marc**

Sur l'intégrabilité uniforme des martingales exponentielles (Martingale theory)

The main result of this paper is the following: Let $X$ be a martingale which is continuous and bounded in $L^1$ (both conditions are essential). Then $X$ is uniformly integrable if and only if $tP\{X^{*}>t\}$ or equivalently $tP\{S(X)>t\}$ tend to $0$ as $t\rightarrow\infty$, where $S(X)$ is the usual square function. The methods (using a good lambda inequality) are close to 1404

Comment: Generalized by Takaoka 3313

Keywords: Exponential martingales, Continuous martingales

Nature: Original

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XIV: 07, 62-75, LNM 784 (1980)

**BARLOW, Martin T.**; **YOR, Marc**

Sur la construction d'une martingale continue de valeur absolue donnée (Martingale theory)

This paper consists of two notes on Gilat's theorem (*Ann. Prob.* **5**, 1977, See also 1358). The problem consists in constructing, given a continuous positive submartingale $Y$, a *continuous * martingale $X$ (possibly on a different space) such that $|X|$ has the same law as $Y$. Let $A$ be the increasing process associated with $Y$; it is necessary for the existence of $X$ that $dA$ should be carried by $\{Y=0\}$. This is shown by the first note (Yor's) to be also sufficient---more precisely, in this case the solutions of Gilat's problem are all continuous. The second note (Barlow's) shows how to construct a Gilat martingale by ``putting a random $\pm$ sign in front of each excursion of $Y$'', a simple intuitive idea and a delicate proof

Keywords: Gilat's theorem

Nature: Original

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XIV: 21, 189-199, LNM 784 (1980)

**YOR, Marc**

Application d'un lemme de Jeulin au grossissement de la filtration brownienne (General theory of processes, Brownian motion)

The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment

Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')

Keywords: Enlargement of filtrations

Nature: Original

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XIV: 22, 200-204, LNM 784 (1980)

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

Construction d'une martingale réelle continue de filtration naturelle donnée (General theory of processes)

It is proved in 1123 that, under a mere separability assumption, any filtration is the natural filtration of some bounded left-continuous increasing process $(A_t)$. If the filtration contains a Brownian motion $(B_t)$, then it is also the natural filtration of the continuous martingale $\int_0^t A_sdB_s$. Therefore, the natural filtration of (finitely or countably) many independent Brownian motions is generated by a single continuous martingale. Explicit constructions are discussed

Nature: Original

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XIV: 38, 343-346, LNM 784 (1980)

**YOR, Marc**

Remarques sur une formule de Paul Lévy (Brownian motion)

Given a two-dimensional Brownian motion $(X_t,Y_t)$, Lévy's area integral formula gives the characteristic function $E[\,\exp(iu\int_0^1 X_s\,dY_s-Y_s\,dX_s)\,\,|\,\, X_0=x, Y_0=y]$. A short proof of this formula is given, and it is shown how to deduce from it the apparently more general $E[\exp(iu\int_0^1 X_sdY_s+iv\int_0^1 Y_sdX_s)\,]$ computed by Berthuet

Keywords: Area integral formula

Nature: Original

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XIV: 48, 496-499, LNM 784 (1980)

**COCOZZA-THIVENT, Christiane**; **YOR, Marc**

Démonstration d'un théorème de F. Knight à l'aide de martingales exponentielles (Martingale theory)

This is a new proof of Knight's theorem that (roughly) finitely many orthogonal continuous local martingales, when separately time-changed into Brownian motions, become independent. A similar theorem for the Poisson case is proved in the same way

Comment: See 518 for an earlier proof

Keywords: Changes of time

Nature: Original

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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: 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: 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

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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

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

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

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

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

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

Nature: Original

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

**YOR, Marc**

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

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

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

Keywords: Martingale inequalities, Domination inequalities

Nature: Original

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

**YOR, Marc**

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

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

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

Keywords: Local times, Hilbert transform, Ito formula

Nature: Original

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XVII: 08, 81-88, LNM 986 (1983)

**LE GALL, Jean-François**; **YOR, Marc**

Sur l'équation stochastique de Tsirelson

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XVII: 09, 89-105, LNM 986 (1983)

**YOR, Marc**

Le drap brownien comme limite en loi des temps locaux linéaires (Brownian motion, Local time, Brownian sheet)

A central limit theorem is obtained for the increments $L^x_t-L^0_t$ of Brownian local times. The limiting process is expressed in terms of a Brownian sheet, independent of the initial Brownian motion

Comment: This type of result is closely related to the Ray-Knight theorems, which describe the law of Brownian local times considered at certain random times. This has been extended first by Rosen in 2533, where Brownian motion is replaced with a symmetric stable process, then by Eisenbaum 2926

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XIX: 29, 332-349, LNM 1123 (1985)

**YOR, Marc**

Compléments aux formules de Tanaka-Rosen (Brownian motion)

Several variants of Rosen's works (*Comm. Math. Phys.* **88** (1983), *Ann. Proba.* **13** (1985), *Ann. Proba.* **14** (1986)) are presented. They yield Tanaka-type formulae for the self-intersection local times of Brownian motion in dimension 2 and beyond, establishing again Varadhan's normalization result (Appendix to *Euclidean quantum field theory*, by K.~Symanzik, in *Local Quantum Theory*, Academic Press, 1969). The methods involve stochastic calculus, which was not needed in 1928

Comment: Examples of further work on this subject, using stochastic calculus or not, are Werner,*Ann. I.H.P.* **29** (1993) who gives many references, Khoshnevisan-Bass, *Ann. I.H.P.* **29** (1993), Rosen-Yor *Ann. Proba.* **19** (1991)

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original proofs

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XIX: 30, 350-365, LNM 1123 (1985)

**YOR, Marc**

Renormalisation et convergence en loi pour des temps locaux d'intersection du mouvement brownien dans ${\bf R}^3$ (Brownian motion)

It is shown that no renormalization à la Varadhan occurs for the self-intersection local times of 3-dimensional Brownian motion; but a weaker result is established: when the point $y\in**R**^3$ tends to $0$, the self-intersection local time at $y$, on the triangle $\{0<s<u\le t\},\ t\ge0$, centered and divided by $(-\log|y|)^{1/2}$, converges in law to a Brownian motion. Several variants of this theorem are established

Comment: This result was used by Le Gall in his work on fluctuations of the Wiener sausage:*Ann. Prob.* **16** (1988). Many results by Rosen have the same flavour

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original

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XX: 34, 532-542, LNM 1204 (1986)

**YOR, Marc**

Précisions sur l'existence et la continuité des temps locaux d'intersection du mouvement brownien dans ${\bf R}^2$

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XX: 35, 543-552, LNM 1204 (1986)

**YOR, Marc**

Sur la représentation comme intégrales stochastiques des temps d'occupation du mouvement brownien dans ${\bf R}^d$ (Brownian motion)

Varadhan's renormalization result (Appendix to*Euclidean quantum field theory*, by K.~Symanzik, in *Local Quantum Theory* consists in centering certain sequences of Brownian functionals and showing $L^2$-convergence. The same results are obtained here by writing these centered functionals as stochastic integrals

Comment: One of mny applications of stochastic calculus to the existence and regularity of self-intersection local times. See Rosen's papers on this topic in general, and page 196 of Le Gall,*École d'Été de Saint-Flour XX*, Springer LNM 1527

Keywords: Local times, Self-intersection, Previsible representation

Nature: Original proofs

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XXI: 15, 230-245, LNM 1247 (1987)

**SONG, Shiqi**; **YOR, Marc**

Inégalités pour les processus self-similaires arrêtés à un temps quelconque

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XXI: 17, 262-269, LNM 1247 (1987)

**AZÉMA, Jacques**; **YOR, Marc**

Interprétation d'un calcul de H. Tanaka en théorie générale des processus

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XXI: 18, 270-275, LNM 1247 (1987)

**BIANE, Philippe**; **LE GALL, Jean-François**; **YOR, Marc**

Un processus qui ressemble au pont brownien

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XXI: 23, 375-403, LNM 1247 (1987)

**CALAIS, J.-Y.**; **YOR, Marc**

Renormalisation et convergence en loi pour certaines intégrales multiples associées au mouvement brownien dans ${\bf R}^d$

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XXII: 23, 217-224, LNM 1321 (1988)

**YOR, Marc**

Remarques sur certaines constructions des mouvements browniens fractionnaires

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XXII: 24, 225-248, LNM 1321 (1988)

**WEINRYB, Sophie**; **YOR, Marc**

Le mouvement brownien de Lévy indexé par ${\bf R}^3$ comme limite centrale de temps locaux d'intersection

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XXII: 34, 454-466, LNM 1321 (1988)

**BIANE, Philippe**; **YOR, Marc**

Sur la loi des temps locaux browniens pris en un temps exponentiel

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XXIII: 07, 88-130, LNM 1372 (1989)

**AZÉMA, Jacques**; **YOR, Marc**

Étude d'une martingale remarquable

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XXIII: 23, 275-293, LNM 1372 (1989)

**BARLOW, Martin T.**; **PITMAN, James W.**; **YOR, Marc**

On Walsh's Brownian motions

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XXIII: 24, 294-314, LNM 1372 (1989)

**BARLOW, Martin T.**; **PITMAN, James W.**; **YOR, Marc**

Une extension multidimensionnelle de la loi de l'arc sinus

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XXIII: 25, 315-323, LNM 1372 (1989)

**DONATI-MARTIN, Catherine**; **YOR, Marc**

Mouvement brownien et inégalité de Hardy dans $L^2$

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XXIV: 14, 210-226, LNM 1426 (1990)

**AZÉMA, Jacques**; **YOR, Marc**

Dérivation par rapport au processus de Bessel

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XXIV: 15, 227-265, LNM 1426 (1990)

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

Filtration des ponts browniens et équations différentielles stochastiques linéaires

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XXV: 23, 284-290, LNM 1485 (1991)

**DUBINS, Lester E.**; **ÉMERY, Michel**; **YOR, Marc**

A continuous martingale in the plane that may spiral away to infinity

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XXVI: 22, 249-306, LNM 1526 (1992)

**AZÉMA, Jacques**; **YOR, Marc**

Sur les zéros des martingales continues

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XXVI: 23, 307-321, LNM 1526 (1992)

**AZÉMA, Jacques**; **MEYER, Paul-André**; **YOR, Marc**

Martingales relatives

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XXVI: 24, 322-347, LNM 1526 (1992)

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

Une décomposition non-canonique du drap brownien (Brownian sheet, Gaussian processes)

In 2415, the authors have introduced a transform of Brownian motion. Here, a similar transform is defined on the Brownian sheet; this transform is shown to be strongly mixing

Comment: This work was motivated by Föllmer's article on Martin boundaries on Wiener space (in*Diffusion processes and related problems in analysis*, vol.~I, Birkhäuser 1990)

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XXVII: 08, 53-77, LNM 1557 (1993)

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

Moyennes mobiles et semimartingales

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XXVII: 14, 122-132, LNM 1557 (1993)

**DUBINS, Lester E.**; **ÉMERY, Michel**; **YOR, Marc**

On the Lévy transformation of Brownian motions and continuous martingales

Comment: An erratum is given in 4421 in Volume XLIV.

Nature: Original

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XXVII: 15, 133-158, LNM 1557 (1993)

**AZÉMA, Jacques**; **JEULIN, Thierry**; **KNIGHT, Frank B.**; **YOR, Marc**

Le théorème d'arrêt en une fin d'ensemble prévisible

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XXVII: 16, 159-172, LNM 1557 (1993)

**ELWORTHY, Kenneth David**; **YOR, Marc**

Conditional expectations for derivatives of certain stochastic flows

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XXVII: 17, 173-176, LNM 1557 (1993)

**WALSH, John B.**; **YOR, Marc**

Some remarks on $A(t,B_t)$

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XXX: 16, 243-254, LNM 1626 (1996)

**AZÉMA, Jacques**; **RAINER, Catherine**; **YOR, Marc**

Une propriété des martingales pures

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XXX: 20, 312-343, LNM 1626 (1996)

**AZÉMA, Jacques**; **JEULIN, Thierry**; **KNIGHT, Frank B.**; **MOKOBODZKI, Gabriel**; **YOR, Marc**

Sur les processus croissants de type injectif

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XXXI: 12, 113-125, LNM 1655 (1997)

**ELWORTHY, Kenneth David**; **LI, Xu-Mei**; **YOR, Marc**

On the tails of the supremum and the quadratic variation of strictly local martingales (Martingale theory)

The asymptotic tails of the current maximum and the quadratic variation of a positive continuous local martingale are compared. Applications to strict local martingales associated with transient diffusions, such as Bessel processes, and remarkable identities for Bessel functions are given

Comment: In discrete time, see the following article 3113. Related results are due to Takaoka 3313

Keywords: Continuous martingales, Local martingales, Quadratic variation, Maximal process

Nature: Original

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XXXI: 27, 272-286, LNM 1655 (1997)

**PITMAN, James W.**; **YOR, Marc**

On the lengths of excursions of some Markov processes

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XXXI: 28, 287-305, LNM 1655 (1997)

**PITMAN, James W.**; **YOR, Marc**

On the relative lengths of excursions derived from a stable subordinator

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XXXI: 29, 306-314, LNM 1655 (1997)

**YOR, Marc**

Some remarks about the joint law of Brownian motion and its supremum (Brownian motion)

Seshadri's identity says that if $S_1$ denotes the maximum of a Brownian motion $B$ on the interval $[0,1]$, the r.v. $2S_1(S_1-B_1)$ is independent of $B_1$ and exponentially distributed. Several variants of this are obtained

Comment: See also 3320

Keywords: Maximal process, Seshadri's identity

Nature: Original

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XXXII: 17, 237-249, LNM 1686 (1998)

**DONEY, R.A.**; **WARREN, Jonathan**; **YOR, Marc**

Perturbed Bessel processes

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XXXII: 19, 264-305, LNM 1686 (1998)

**BARLOW, Martin T.**; **ÉMERY, Michel**; **KNIGHT, Frank B.**; **SONG, Shiqi**; **YOR, Marc**

Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)

Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA**7**, 1997). Using his methods, the result is extended to spider martingales. A conjecture of M. Barlow is also proved: if $L$ is an honest time in a (possibly multidimensional) Brownian filtration, then ${\cal F}_{L+}$ is generated by ${\cal F}_{L}$ and at most one event. Last, it is shown that a Walsh's Brownian motion can live in the filtration generated by another Walsh's Brownian motion only if the former is obtained from the latter by aggregating rays

Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,*Astérisque* **282** (2002). A simplified proof of Barlow's conjecture is given in 3304. For more on Théorème 1 (Slutsky's lemma), see 3221 and 3325

Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma

Nature: New exposition of known results, Original additions

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XXXII: 20, 306-312, LNM 1686 (1998)

**ÉMERY, Michel**; **YOR, Marc**

Sur un théorème de Tsirelson relatif à des mouvements browniens corrélés et à la nullité de certains temps locaux

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XXXII: 22, 316-327, LNM 1686 (1998)

**AZÉMA, Jacques**; **JEULIN, Thierry**; **KNIGHT, Frank B.**; **YOR, Marc**

Quelques calculs de compensateurs impliquant l'injectivité de certains processus croissants

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XXXII: 23, 328-342, LNM 1686 (1998)

**WARREN, Jonathan**; **YOR, Marc**

The Brownian burglar: conditioning Brownian motion by its local time process

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XXXIV: 20, 417-431, LNM 1729 (2000)

**VOSTRIKOVA, Lioudmilla**; **YOR, Marc**

Some invariance properties (of the laws) of Ocone's martingales

Comment: One page of this article is misplaced: Page 421 should not be read just after page 420, but after page 425

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XXXV: 23, 334-347, LNM 1755 (2001)

**CHAUMONT, Loïc**; **HOBSON, David G.**; **YOR, Marc**

Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes

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XXXVIII: 05, 42-69, LNM 1857 (2005)

**NGUYEN-NGOC, Laurent**; **YOR, Marc**

Some martingales associated to reflected Lévy processes

XXXIX: 08, 157-170, LNM 1874 (2006)

**MADAN, Dilip B.**; **YOR, Marc**

Ito's integrated formula for strict local martingales

XXXIX: 15, 305-336, LNM 1874 (2006)

**ROYNETTE, Bernard**; **VALLOIS, Pierre**; **YOR, Marc**

Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère

XXXIX: 16, 337-356, LNM 1874 (2006)

**GALLARDO, Léonard**; **YOR, Marc**

Some remarkable properties of the Dunkl martingales

XL: 14, 265-285, LNM 1899 (2007)

**SALMINEN, Paavo**; **YOR, Marc**

Tanaka formulae for symmetric Lévy processes

XLII: 08, 187-227, LNM 1978 (2009)

**YANO, Kouji**; **YANO, Yuko**; **YOR, Marc**

On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes (Theory of Lévy processes)

Nature: Original

XLIII: 19, 437-439, LNM 2006 (2011)

**BAKER, David**; **YOR, Marc**

On martingales with given marginals and the scaling property (Martingale theory, Theory of Brownian motion)

Nature: Original

XLIII: 20, 441-449, LNM 2006 (2011)

**BAKER, David**; **DONATI-MARTIN, Catherine**; **YOR, Marc**

A sequence of Albin type continuous martingales with Brownian marginals and scaling (Martingale theory)

Keywords: Martingales, Brownian marginals

Nature: Original

XLIII: 21, 451-503, LNM 2006 (2011)

**HIRSCH, Francis**; **PROFETA, Christophe**; **ROYNETTE, Bernard**; **YOR, Marc**

Constructing self-similar martingales via two Skorokhod embeddings (Martingale theory)

Keywords: Skorokhod embeddings, Hardy-Littlewood functions, Convex order, Schauder fixed point theorem, Self-similar martingales, Karamata's representation theorem

Nature: Original

XLIV: 21, 467-467, LNM 2046 (2012)

**ÉMERY, Michel**; **YOR, Marc**

Erratum to Séminaire XXVII

Comment: This is an erratum to 2714.

Keywords: Brownian motion, Continuous martingale

Nature: Correction

XLVI: 14, 359-375, LNM 2123 (2014)

**ROSENBAUM, Mathieu**; **YOR, Marc**

On the law of a triplet associated with the pseudo-Brownian bridge (Theory of Brownian motion)

This article gives a remarkable identity in law which relates the Brownian motion, its local time, and the the inverse of its local time

Keywords: Brownian motion, pseudo-Brownian bridge, Bessel process, local time, hitting times, scaling, uniform sampling, Mellin transform

Nature: Original

XLVII: 03, 1-15, LNM 2137 (2015)

**SALMINEN, Paavo**; **YEN, Ju-Yi**; **YOR, Marc**

Integral Representations of Certain Measures in the One-Dimensional Diffusions Excursion Theory

Nature: Original

Sur la théorie de la prédiction, et le problème de décomposition des tribus ${\cal F}^{\circ}_{t+}$ (General theory of processes)

This paper contains another version of Knight's theory (preceding paper 1007) for cadlag process instead of measurable processes. These results then are applied to the pathology of germ fields: a natural measurability conjecture does not hold, and an example is given of a process $X_t$ such that its natural $\sigma$-field ${\cal F}_{1+}$ is not generated by ${\cal F}_{1}$ and the germ-field at $0$ of the process $(X_{1+s})$

Comment: On the pathology of germ fields, see H. von Weizsäcker,

Keywords: Prediction theory, Germ fields

Nature: Original

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X: 22, 481-500, LNM 511 (1976)

Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (Martingale theory, Stochastic calculus)

This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of ``exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up

Keywords: Optional stochastic integrals, Stochastic exponentials, Lévy systems

Nature: Original

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XI: 14, 257-297, LNM 581 (1977)

Sur les théories du filtrage et de la prédiction (General theory of processes, Markov processes)

To be completed

Comment: MR 57, 10801

Keywords: Filtering theory, Prediction theory

Nature: Original

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XI: 34, 493-501, LNM 581 (1977)

A propos d'un lemme de Ch. Yoeurp (General theory of processes, Martingale theory)

Yoeurp's lemma is the following: if $A$ is a previsible process of bounded variation, its square bracket $[A,L]$ with any local martingale $L$ is a local martingale. This useful result was not easily accessible, thus a complete proof is given, with several new applications---in particular, this characterizes previsible processes of bounded variation among semimartingales

Keywords: Yoeurp's lemma, Square bracket

Nature: Original

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XI: 35, 502-517, LNM 581 (1977)

Remarques sur la représentation des martingales comme intégrales stochastiques (Martingale theory)

The main result on the relation between the previsible representation property of a set of local martingales and the extremality of their joint law appeared in a celebrated paper of Jacod-Yor,

Comment: This is an intermediate paper between the Jacod-Yor results and the definitive version of previsible representation, using the theorem of Douglas, for which see 1221

Keywords: Previsible representation, Extreme points, Independent increments, Lévy processes

Nature: Original

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XI: 36, 518-528, LNM 581 (1977)

Sur quelques approximations d'intégrales stochastiques (Martingale theory)

The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form

Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe

Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals

Nature: Original

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XII: 09, 61-69, LNM 649 (1978)

Grossissement d'une filtration et semi-martingales~: théorèmes généraux (General theory of processes)

Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called

Comment: This result was independently discovered by Barlow,

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 11, 78-97, LNM 649 (1978)

Grossissement d'une filtration et semi-martingales~: Formules explicites (General theory of processes)

This contains very substantial improvements on 1209, namely, the explicit computation of the characteristics of the semimartingales involved

Comment: For additional results on enlargements, see the two Lecture Notes volumes

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 12, 98-113, LNM 649 (1978)

Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)

The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)

Keywords: Hardy spaces, $BMO$

Nature: Original

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XII: 21, 265-309, LNM 649 (1978)

Sous-espaces denses dans $L^1$ ou $H^1$ et représentation des martingales (Martingale theory)

This paper was a considerable step in the study of the general martingale problem, i.e., of the set ${\cal P}$ of all laws on a filtered measurable space under which a given set ${\cal N}$ of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R.G. Douglas (

Comment: The second named author's contribution concerns only the appendix on homogeneous martingales

Keywords: Previsible representation, Douglas theorem, Extremal laws

Nature: Original

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XII: 35, 482-488, LNM 649 (1978)

Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (Measure theory)

Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability

Comment: The subject is discussed further in 1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the

Keywords: Kernels, Radon-Nikodym theorem

Nature: Original

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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XIV: 06, 53-61, LNM 784 (1980)

Sur l'intégrabilité uniforme des martingales exponentielles (Martingale theory)

The main result of this paper is the following: Let $X$ be a martingale which is continuous and bounded in $L^1$ (both conditions are essential). Then $X$ is uniformly integrable if and only if $tP\{X^{*}>t\}$ or equivalently $tP\{S(X)>t\}$ tend to $0$ as $t\rightarrow\infty$, where $S(X)$ is the usual square function. The methods (using a good lambda inequality) are close to 1404

Comment: Generalized by Takaoka 3313

Keywords: Exponential martingales, Continuous martingales

Nature: Original

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XIV: 07, 62-75, LNM 784 (1980)

Sur la construction d'une martingale continue de valeur absolue donnée (Martingale theory)

This paper consists of two notes on Gilat's theorem (

Keywords: Gilat's theorem

Nature: Original

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XIV: 21, 189-199, LNM 784 (1980)

Application d'un lemme de Jeulin au grossissement de la filtration brownienne (General theory of processes, Brownian motion)

The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment

Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')

Keywords: Enlargement of filtrations

Nature: Original

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XIV: 22, 200-204, LNM 784 (1980)

Construction d'une martingale réelle continue de filtration naturelle donnée (General theory of processes)

It is proved in 1123 that, under a mere separability assumption, any filtration is the natural filtration of some bounded left-continuous increasing process $(A_t)$. If the filtration contains a Brownian motion $(B_t)$, then it is also the natural filtration of the continuous martingale $\int_0^t A_sdB_s$. Therefore, the natural filtration of (finitely or countably) many independent Brownian motions is generated by a single continuous martingale. Explicit constructions are discussed

Nature: Original

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XIV: 38, 343-346, LNM 784 (1980)

Remarques sur une formule de Paul Lévy (Brownian motion)

Given a two-dimensional Brownian motion $(X_t,Y_t)$, Lévy's area integral formula gives the characteristic function $E[\,\exp(iu\int_0^1 X_s\,dY_s-Y_s\,dX_s)\,\,|\,\, X_0=x, Y_0=y]$. A short proof of this formula is given, and it is shown how to deduce from it the apparently more general $E[\exp(iu\int_0^1 X_sdY_s+iv\int_0^1 Y_sdX_s)\,]$ computed by Berthuet

Keywords: Area integral formula

Nature: Original

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XIV: 48, 496-499, LNM 784 (1980)

Démonstration d'un théorème de F. Knight à l'aide de martingales exponentielles (Martingale theory)

This is a new proof of Knight's theorem that (roughly) finitely many orthogonal continuous local martingales, when separately time-changed into Brownian motions, become independent. A similar theorem for the Poisson case is proved in the same way

Comment: See 518 for an earlier proof

Keywords: Changes of time

Nature: Original

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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

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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

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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

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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

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

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

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

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

Nature: Original

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

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

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

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

Keywords: Martingale inequalities, Domination inequalities

Nature: Original

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

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

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

Comment: For a modern version with references see Yor,

Keywords: Local times, Hilbert transform, Ito formula

Nature: Original

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XVII: 08, 81-88, LNM 986 (1983)

Sur l'équation stochastique de Tsirelson

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XVII: 09, 89-105, LNM 986 (1983)

Le drap brownien comme limite en loi des temps locaux linéaires (Brownian motion, Local time, Brownian sheet)

A central limit theorem is obtained for the increments $L^x_t-L^0_t$ of Brownian local times. The limiting process is expressed in terms of a Brownian sheet, independent of the initial Brownian motion

Comment: This type of result is closely related to the Ray-Knight theorems, which describe the law of Brownian local times considered at certain random times. This has been extended first by Rosen in 2533, where Brownian motion is replaced with a symmetric stable process, then by Eisenbaum 2926

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XIX: 29, 332-349, LNM 1123 (1985)

Compléments aux formules de Tanaka-Rosen (Brownian motion)

Several variants of Rosen's works (

Comment: Examples of further work on this subject, using stochastic calculus or not, are Werner,

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original proofs

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XIX: 30, 350-365, LNM 1123 (1985)

Renormalisation et convergence en loi pour des temps locaux d'intersection du mouvement brownien dans ${\bf R}^3$ (Brownian motion)

It is shown that no renormalization à la Varadhan occurs for the self-intersection local times of 3-dimensional Brownian motion; but a weaker result is established: when the point $y\in

Comment: This result was used by Le Gall in his work on fluctuations of the Wiener sausage:

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original

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XX: 34, 532-542, LNM 1204 (1986)

Précisions sur l'existence et la continuité des temps locaux d'intersection du mouvement brownien dans ${\bf R}^2$

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XX: 35, 543-552, LNM 1204 (1986)

Sur la représentation comme intégrales stochastiques des temps d'occupation du mouvement brownien dans ${\bf R}^d$ (Brownian motion)

Varadhan's renormalization result (Appendix to

Comment: One of mny applications of stochastic calculus to the existence and regularity of self-intersection local times. See Rosen's papers on this topic in general, and page 196 of Le Gall,

Keywords: Local times, Self-intersection, Previsible representation

Nature: Original proofs

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XXI: 15, 230-245, LNM 1247 (1987)

Inégalités pour les processus self-similaires arrêtés à un temps quelconque

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XXI: 17, 262-269, LNM 1247 (1987)

Interprétation d'un calcul de H. Tanaka en théorie générale des processus

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XXI: 18, 270-275, LNM 1247 (1987)

Un processus qui ressemble au pont brownien

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XXI: 23, 375-403, LNM 1247 (1987)

Renormalisation et convergence en loi pour certaines intégrales multiples associées au mouvement brownien dans ${\bf R}^d$

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XXII: 23, 217-224, LNM 1321 (1988)

Remarques sur certaines constructions des mouvements browniens fractionnaires

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XXII: 24, 225-248, LNM 1321 (1988)

Le mouvement brownien de Lévy indexé par ${\bf R}^3$ comme limite centrale de temps locaux d'intersection

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XXII: 34, 454-466, LNM 1321 (1988)

Sur la loi des temps locaux browniens pris en un temps exponentiel

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XXIII: 07, 88-130, LNM 1372 (1989)

Étude d'une martingale remarquable

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XXIII: 23, 275-293, LNM 1372 (1989)

On Walsh's Brownian motions

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XXIII: 24, 294-314, LNM 1372 (1989)

Une extension multidimensionnelle de la loi de l'arc sinus

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XXIII: 25, 315-323, LNM 1372 (1989)

Mouvement brownien et inégalité de Hardy dans $L^2$

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XXIV: 14, 210-226, LNM 1426 (1990)

Dérivation par rapport au processus de Bessel

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XXIV: 15, 227-265, LNM 1426 (1990)

Filtration des ponts browniens et équations différentielles stochastiques linéaires

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XXV: 23, 284-290, LNM 1485 (1991)

A continuous martingale in the plane that may spiral away to infinity

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XXVI: 22, 249-306, LNM 1526 (1992)

Sur les zéros des martingales continues

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XXVI: 23, 307-321, LNM 1526 (1992)

Martingales relatives

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XXVI: 24, 322-347, LNM 1526 (1992)

Une décomposition non-canonique du drap brownien (Brownian sheet, Gaussian processes)

In 2415, the authors have introduced a transform of Brownian motion. Here, a similar transform is defined on the Brownian sheet; this transform is shown to be strongly mixing

Comment: This work was motivated by Föllmer's article on Martin boundaries on Wiener space (in

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XXVII: 08, 53-77, LNM 1557 (1993)

Moyennes mobiles et semimartingales

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XXVII: 14, 122-132, LNM 1557 (1993)

On the Lévy transformation of Brownian motions and continuous martingales

Comment: An erratum is given in 4421 in Volume XLIV.

Nature: Original

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XXVII: 15, 133-158, LNM 1557 (1993)

Le théorème d'arrêt en une fin d'ensemble prévisible

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XXVII: 16, 159-172, LNM 1557 (1993)

Conditional expectations for derivatives of certain stochastic flows

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XXVII: 17, 173-176, LNM 1557 (1993)

Some remarks on $A(t,B_t)$

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XXX: 16, 243-254, LNM 1626 (1996)

Une propriété des martingales pures

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XXX: 20, 312-343, LNM 1626 (1996)

Sur les processus croissants de type injectif

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XXXI: 12, 113-125, LNM 1655 (1997)

On the tails of the supremum and the quadratic variation of strictly local martingales (Martingale theory)

The asymptotic tails of the current maximum and the quadratic variation of a positive continuous local martingale are compared. Applications to strict local martingales associated with transient diffusions, such as Bessel processes, and remarkable identities for Bessel functions are given

Comment: In discrete time, see the following article 3113. Related results are due to Takaoka 3313

Keywords: Continuous martingales, Local martingales, Quadratic variation, Maximal process

Nature: Original

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XXXI: 27, 272-286, LNM 1655 (1997)

On the lengths of excursions of some Markov processes

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XXXI: 28, 287-305, LNM 1655 (1997)

On the relative lengths of excursions derived from a stable subordinator

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XXXI: 29, 306-314, LNM 1655 (1997)

Some remarks about the joint law of Brownian motion and its supremum (Brownian motion)

Seshadri's identity says that if $S_1$ denotes the maximum of a Brownian motion $B$ on the interval $[0,1]$, the r.v. $2S_1(S_1-B_1)$ is independent of $B_1$ and exponentially distributed. Several variants of this are obtained

Comment: See also 3320

Keywords: Maximal process, Seshadri's identity

Nature: Original

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XXXII: 17, 237-249, LNM 1686 (1998)

Perturbed Bessel processes

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XXXII: 19, 264-305, LNM 1686 (1998)

Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)

Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA

Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,

Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma

Nature: New exposition of known results, Original additions

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XXXII: 20, 306-312, LNM 1686 (1998)

Sur un théorème de Tsirelson relatif à des mouvements browniens corrélés et à la nullité de certains temps locaux

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XXXII: 22, 316-327, LNM 1686 (1998)

Quelques calculs de compensateurs impliquant l'injectivité de certains processus croissants

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XXXII: 23, 328-342, LNM 1686 (1998)

The Brownian burglar: conditioning Brownian motion by its local time process

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XXXIV: 20, 417-431, LNM 1729 (2000)

Some invariance properties (of the laws) of Ocone's martingales

Comment: One page of this article is misplaced: Page 421 should not be read just after page 420, but after page 425

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XXXV: 23, 334-347, LNM 1755 (2001)

Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes

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XXXVIII: 05, 42-69, LNM 1857 (2005)

Some martingales associated to reflected Lévy processes

XXXIX: 08, 157-170, LNM 1874 (2006)

Ito's integrated formula for strict local martingales

XXXIX: 15, 305-336, LNM 1874 (2006)

Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère

XXXIX: 16, 337-356, LNM 1874 (2006)

Some remarkable properties of the Dunkl martingales

XL: 14, 265-285, LNM 1899 (2007)

Tanaka formulae for symmetric Lévy processes

XLII: 08, 187-227, LNM 1978 (2009)

On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes (Theory of Lévy processes)

Nature: Original

XLIII: 19, 437-439, LNM 2006 (2011)

On martingales with given marginals and the scaling property (Martingale theory, Theory of Brownian motion)

Nature: Original

XLIII: 20, 441-449, LNM 2006 (2011)

A sequence of Albin type continuous martingales with Brownian marginals and scaling (Martingale theory)

Keywords: Martingales, Brownian marginals

Nature: Original

XLIII: 21, 451-503, LNM 2006 (2011)

Constructing self-similar martingales via two Skorokhod embeddings (Martingale theory)

Keywords: Skorokhod embeddings, Hardy-Littlewood functions, Convex order, Schauder fixed point theorem, Self-similar martingales, Karamata's representation theorem

Nature: Original

XLIV: 21, 467-467, LNM 2046 (2012)

Erratum to Séminaire XXVII

Comment: This is an erratum to 2714.

Keywords: Brownian motion, Continuous martingale

Nature: Correction

XLVI: 14, 359-375, LNM 2123 (2014)

On the law of a triplet associated with the pseudo-Brownian bridge (Theory of Brownian motion)

This article gives a remarkable identity in law which relates the Brownian motion, its local time, and the the inverse of its local time

Keywords: Brownian motion, pseudo-Brownian bridge, Bessel process, local time, hitting times, scaling, uniform sampling, Mellin transform

Nature: Original

XLVII: 03, 1-15, LNM 2137 (2015)

Integral Representations of Certain Measures in the One-Dimensional Diffusions Excursion Theory

Nature: Original