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18 matches found
VIII: 05, 25-26, LNM 381 (1974)
DELLACHERIE, Claude
Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (General theory of processes)
This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems
Comment: A gap in the proof is filled in 928 and 2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor, Z. für W-theorie, 38, 1977 and Yor 1221. For another approach to the restricted case considered here, see Ruiz de Chavez 1821. The previsible representation property of Brownian motion and compensated Poisson process was know by Itô; it is a consequence of the (stronger) chaotic representation property, established by Wiener in 1938. The converse was also known by Itô: among the martingales which are also Lévy processes, only Brownian motions and compensated Poisson processes have the previsible representation property
Keywords: Brownian motion, Poisson processes, Previsible representation
Nature: Original
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IX: 06, 226-236, LNM 465 (1975)
CHOU, Ching Sung; MEYER, Paul-André
Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels (General theory of processes)
Dellacherie has studied in 405 the filtration generated by a point process with one single jump. His study is extended here to the filtration generated by a discrete point process. It is shown in particular how to construct a martingale which has the previsible representation property
Comment: In spite or because of its simplicity, this paper has become a standard reference in the field. For a general account of the subject, see He-Wang-Yan, Semimartingale Theory and Stochastic Calculus, CRC~Press 1992
Keywords: Point processes, Previsible representation
Nature: Original
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IX: 28, 494-494, LNM 465 (1975)
DELLACHERIE, Claude
Correction à Intégrales stochastiques par rapport...'' (General theory of processes)
This paper completes a gap in the simple proof of the previsible representation property of the Wiener process, given by Dellacherie 805
Comment: Another way of filling this gap is given by Ruiz de Chavez 1821. The same gap for the Poisson process is corrected in 2002
Keywords: Previsible representation
Nature: Original
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X: 01, 1-18, LNM 511 (1976)
BRÉMAUD, Pierre
La méthode des semi-martingales en filtrage quand l'observation est un processus ponctuel marqué (Martingale theory, Point processes)
This paper discusses martingale methods (as developed by Jacod, Z. für W-theorie, 31, 1975) in the filtering theory of point processes
Comment: The author has greatly developed this topic in his book Poisson Processes and Queues, Springer 1981
Keywords: Point processes, Previsible representation, Filtering theory
Nature: Original
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X: 03, 24-39, LNM 511 (1976)
JACOD, Jean; MÉMIN, Jean
Un théorème de représentation des martingales pour les ensembles régénératifs (Martingale theory, Markov processes, Stochastic calculus)
The natural filtration of a regenerative set $M$ is that of the corresponding age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration
Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in 1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second Azéma's martingale'' (not the well known one which has the chaotic representation property)
Keywords: Regenerative sets, Renewal theory, Stochastic integrals, Previsible representation
Nature: Original
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X: 17, 245-400, LNM 511 (1976)
MEYER, Paul-André
Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)
This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$
Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books
Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem
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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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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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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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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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XV: 42, 618-626, LNM 850 (1981)
ITMI, Mhamed
Processus ponctuels marqués stochastiques. Représentation des martingales et filtration naturelle quasicontinue à gauche (General theory of processes)
This paper contains a study of the filtration generated by a point process (multivariate: it takes values in a Polish space), and in particular of its quasi-left continuity, and previsible representation
Keywords: Point processes, Previsible representation
Nature: Original
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XVI: 18, 219-220, LNM 920 (1982)
STRICKER, Christophe
Les intervalles de constance de $\langle X,X\rangle$ (Martingale theory, Stochastic calculus)
For a continuous (local) martingale $X$, the constancy intervals of $X$ and $<X,X>$ are exactly the same. What about general local martingales? It is proved that $X$ is constant on the constancy intervals of $<X,X>$, and the converse holds if $X$ has the previsible representation property
Nature: Original
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XVI: 30, 348-354, LNM 920 (1982)
HE, Sheng-Wu; WANG, Jia-Gang
The total continuity of natural filtrations (General theory of processes)
Total continuity of a filtration ${\cal F}$ means that ${\cal F}_T={\cal F}_{T-}$ at every stopping time $T$, not necessarily previsible. It is shown that the filtration of a Lévy process without fixed discontinuities is totally continuous if and only if the jump size is a deterministic function of the jump time. Similarly, the natural filtration of a quasi-left continuous jump process is totally continuous if and only if the size of the $n$-th jump is a deterministic function of the jump times up to the $n$-th. It is shown that under the usual (here called strong'') previsible representation property, quasi-left continuity of the filtration implies total continuity
Keywords: Filtrations, Independent increments, Previsible representation, Total continuity, Lévy processes
Nature: Original
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XVII: 21, 194-197, LNM 986 (1983)
PRICE, Gareth C.; WILLIAMS, David
Rolling with `slipping': I (Stochastic calculus, Stochastic differential geometry)
If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$
Keywords: Brownian motion in a manifold, Previsible representation
Nature: Original
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XX: 02, 28-29, LNM 1204 (1986)
FAGNOLA, Franco; LETTA, Giorgio
Sur la représentation intégrale des martingales du processus de Poisson (Stochastic calculus, Point processes)
Dellacherie gave in 805 a proof by stochastic calculus of the previsible representation property for the Wiener and Poisson processes. A gap in this proof is filled in 928 for Brownian motion and here for Poisson processes
Keywords: Stochastic integrals, Previsible representation, Poisson processes
Nature: Correction
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XX: 31, 465-502, LNM 1204 (1986)
McGILL, Paul
Integral representation of martingales in the Brownian excursion filtration (Brownian motion, Stochastic calculus)
An integral representation is obtained of all square integrable martingales in the filtration $({\cal E}^x,\ x\inR)$, where ${\cal E}^x$ denotes the Brownian excursion $\sigma$-field below $x$ introduced by D. Williams 1343, who also showed that every $({\cal E}^x)$ martingale is continuous
Comment: Another filtration $(\tilde{\cal E}^x,\ x\inR)$ of Brownian excursions below $x$ has been proposed by Azéma; the structure of martingales is quite diffferent: they are discontinuous. See Y. Hu's thesis (Paris VI, 1996), and chap.~16 of Yor, Some Aspects of Brownian Motion, Part~II, Birkhäuser, 1997
Keywords: Previsible representation, Martingales, Filtrations
Nature: Original
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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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