V: 27, 278-282, LNM 191 (1971)
SAM LAZARO, José de;
MEYER, Paul-André
Une remarque sur le flot du mouvement brownien (
Brownian motion,
Ergodic theory)
It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense
Comment: See Sam Lazaro-Meyer,
Z. für W-theorie, 18, 1971
Keywords: Brownian motion,
Wiener chaos,
Screw-linesNature: Original
Retrieve article from Numdam
VIII: 02, 11-19, LNM 381 (1974)
BRETAGNOLLE, Jean
Une remarque sur le problème de Skorohod (
Brownian motion)
The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see
516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
VIII: 10, 134-149, LNM 381 (1974)
KNIGHT, Frank B.
Existence of small oscillations at zeros of brownian motion (
Brownian motion)
The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an ``iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros
Comment: This result may be compared to Kahane,
C.R. Acad. Sci. 248, 1974
Keywords: Law of the iterated logarithm,
Local timesNature: Original Retrieve article from Numdam
X: 02, 19-23, LNM 511 (1976)
CHACON, Rafael V.;
WALSH, John B.
One-dimensional potential imbedding (
Brownian motion)
The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool
Comment: The construction is related to that of Dubins (see
516). In this volume
1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
X: 15, 235-239, LNM 511 (1976)
WILLIAMS, David
On a stopped Brownian motion formula of H.M.~Taylor (
Brownian motion)
This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given
Comment: For the original proof of Taylor see
Ann. Prob. 3, 1975. For modern references, we should ask Yor
Keywords: Stopping times,
Local times,
Ray-Knight theorems,
Cameron-Martin formulaNature: Original Retrieve article from Numdam
XII: 31, 428-445, LNM 649 (1978)
KNIGHT, Frank B.
On the sojourn times of killed Brownian motion (
Brownian motion)
To be completed
Keywords: Sojourn times,
Laplace transformsNature: Original Retrieve article from Numdam
XII: 57, 763-769, LNM 649 (1978)
MEYER, Paul-André
La formule d'Ito pour le mouvement brownien, d'après Brosamler (
Brownian motion,
Stochastic calculus)
This paper presents the results of a paper by Brosamler (
Trans. Amer. Math. Soc. 149, 1970) on the Ito formula $f(B_t)=...$ for $n$-dimensional Brownian motion, under the weakest possible assumptions: namely up to the first exit time from an open set $W$ and assuming only that $f$ is locally in $L^1$ in $W$, and its Laplacian in the sense of distributions is a measure in $W$
Keywords: Ito formulaNature: Exposition Retrieve article from Numdam
XIII: 06, 90-115, LNM 721 (1979)
AZÉMA, Jacques;
YOR, Marc
Une solution simple au problème de Skorokhod (
Brownian motion)
An explicit solution is given to Skorohod's problem: given a distribution $\mu$ with mean $0$ and finite second moment $\sigma^2$, find a (non randomized) stopping time $T$ of a Brownian motion $(X_t)$ such that $X_T$ has the distribution $\mu$ and $E[T]=\sigma^2$. It is shown that if $S_t$ is the one-sided supremum of $X$ at time $t$, $T=\inf\{t:S_t\ge\psi(X_t)\}$ solves the problem, where $\psi(x)$ is the barycenter of $\mu$ restricted to $[x,\infty[$. The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray-Knight theorem on local times
Comment: The subject is further investigated in
1356 and
1441. See also
1515. A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XIII: 36, 427-440, LNM 721 (1979)
YOR, Marc
Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (
Brownian motion)
The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property
Comment: Additional results on the same subject appear in
1545 and in Malric
Ann. Inst. H. Poincaré 26 (1990)
Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XIII: 43, 490-494, LNM 721 (1979)
WILLIAMS, David
Conditional excursion theory (
Brownian motion,
Markov processes)
To be completed
Keywords: ExcursionsNature: Original Retrieve article from Numdam
XIII: 45, 521-532, LNM 721 (1979)
JEULIN, Thierry
Un théorème de J.W. Pitman (
Brownian motion,
Diffusion theory)
This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$
Keywords: Bessel processesNature: New proof of known results Retrieve article from Numdam
XIII: 56, 625-633, LNM 721 (1979)
AZÉMA, Jacques;
YOR, Marc
Le problème de Skorokhod~: compléments à l'exposé précédent (
Brownian motion)
What the title calls ``the preceding talk'' is
1306. The method is extended to (centered) measures possessing a moment of order one instead of two, preserving the uniform integrability of the stopped martingale
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XIII: 59, 646-646, LNM 721 (1979)
BARLOW, Martin T.
On the left endpoints of Brownian excursions (
Brownian motion,
Excursion theory)
It is shown that no expansion of the Brownian filtration can be found such that $B_t$ remains a semimartingale, and the set of left endpoints of Brownian excursions becomes optional
Keywords: Progressive setsNature: Original Retrieve article from Numdam
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 filtrationsNature: Original Retrieve article from Numdam
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 formulaNature: Original Retrieve article from Numdam
XIV: 40, 357-391, LNM 784 (1980)
FALKNER, Neil
On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times (
Brownian motion,
Potential theory)
The problem discussed here is the Skorohod representation of a measure $\nu$ as the distribution of $B_T$, where $(B_t)$ is Brownian motion in $
R^n$ with the initial measure $\mu$, and $T$ is a
non-randomized stopping time. The conditions given are sufficient in all cases, necessary if $\mu$ does not charge polar sets
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XIV: 41, 392-396, LNM 784 (1980)
PIERRE, Michel
Le problème de Skorohod~: une remarque sur la démonstration d'Azéma-Yor (
Brownian motion)
This is an addition to
1306 and
1356, showing how the proof can be reduced to that of a regular case, where it becomes simpler
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XV: 01, 1-5, LNM 850 (1981)
FERNIQUE, Xavier
Sur les lois de certaines intégrales associées à des mouvements browniens (
Brownian motion)
Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)
Keywords: Iterated stochastic integralsNature: Original Retrieve article from Numdam
XV: 12, 189-190, LNM 850 (1981)
BARLOW, Martin T.
On Brownian local time (
Brownian motion)
Let $(L^a_t)$ be the standard (jointly continuous) version of Brownian local times. Perkins has shown that for fixed $t$ $a\mapsto L^a_t$ is a semimartingale relative to the excusrion fields. An example is given of a stopping time $T$ for which $a\mapsto L^a_T$ is not a semimartingale
Keywords: Local timesNature: Original Retrieve article from Numdam
XV: 14, 206-209, LNM 850 (1981)
McGILL, Paul
A direct proof of the Ray-Knight theorem (
Brownian motion)
The (first) Ray-Knight theorem describes the law of the process $(L_T^{1-a})_{0\le a\le 1}$ where $(L^a_t)$ is the family of local times of Brownian motion starting from $0$ and $T$ is the hitting time of $1$. A direct proof is given indeed. It is reproduced in Revuz-Yor,
Continuous Martingales and Brownian Motion, Chapter XI, exercice (2.7)
Keywords: Local times,
Ray-Knight theorems,
Bessel processesNature: New proof of known results Retrieve article from Numdam
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
1516Keywords: Explicit laws,
Occupation times,
Enlargement of filtrations,
Williams decompositionNature: Original Retrieve article from Numdam
XV: 16, 227-250, LNM 850 (1981)
ROGERS, L.C.G.
Williams' characterization of the Brownian excursion law: proof and applications (
Brownian motion)
In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams
Diffusions, Markov Processes and Martingales, Wiley 1979, II.67), and it collects a number of applications, including the Azéma-Yor approach to Skorohod's imbedding theorem (
1306)
Keywords: Excursions,
Explicit laws,
Bessel processes,
Skorohod imbeddingNature: Original Retrieve article from Numdam
XV: 29, 399-412, LNM 850 (1981)
YOEURP, Chantha
Sur la dérivation stochastique au sens de Davis (
Stochastic calculus,
Brownian motion)
The problem is that of estimating $H_t$ from a knowledge of the process $\int H_sdX_s$, where $X_t$ is a continuous (semi)martingale, as a limit of ratios of the form $\int_t^{t+\alpha} H_sdX_s/(X_{t+\alpha}-X_t)$, or replacing $t+\alpha$ by suitable stopping times
Comment: The problem was suggested and partially solved by Mark H.A. Davis (
Teor. Ver. Prim.,
20, 1975, 887--892). See also
1428Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XV: 45, 643-668, LNM 850 (1981)
AUERHAN, J.;
LÉPINGLE, Dominique
Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (
General theory of processes,
Brownian motion,
Martingale theory)
This is a sequel to
1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in
1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given
Comment: Further results are given by Malric
Ann. Inst. H. Poincaré 26 (1990)
Nature: Original Retrieve article from Numdam
XV: 46, 669-670, LNM 850 (1981)
LÉPINGLE, Dominique
Une remarque sur les lois de certains temps d'atteinte (
Brownian motion)
Let $T$ be the exit time of the interval $[-d,c]$ for a Brownian motion starting at $0$. A classical formula giving the Laplace transform of the law of $T$ can be extended by analytical continuation to the positive axis. It is shown here that this extension has a purely probabilistic proof. The same method gives two other formulas
Keywords: Exit time from an intervalNature: New proof of known results Retrieve article from Numdam
XVI: 15, 209-211, LNM 920 (1982)
BARLOW, Martin T.
$L(B_t,t)$ is not a semimartingale (
Brownian motion)
The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale
Keywords: Local times,
SemimartingalesNature: Original Retrieve article from Numdam
XVI: 20, 234-237, LNM 920 (1982)
YOEURP, Chantha
Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (
Brownian motion,
Stochastic calculus)
A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$
Comment: See
1023,
1321Keywords: Multiplicative decomposition,
Change of variable formula,
Local timesNature: Original Retrieve article from Numdam
XVI: 21, 238-247, LNM 920 (1982)
YOR, Marc
Sur la transformée de Hilbert des temps locaux browniens et une extension de la formule d'Itô (
Brownian motion)
This paper is about the application to the function $(x-a)\log|x-a|-(x-a)$ (whose second derivative is $1/x-a$) of the Ito-Tanaka formula; the last term then involves a formal Hilbert transform $\tilde L^a_t$ of the local time process $L^a_t$. Such processes had been defined by Ito and McKean, and studied by Yamada as examples of Fukushima's ``additive functionals of zero energy''. Here it is proved, as a consequence of a general theorem, that this process has a jointly continuous version---more precisely, Hölder continuous of all orders $<1/2$ in $a$ and in $t$
Comment: For a modern version with references see Yor,
Some Aspects of Brownian Motion II, Birkhäuser 1997
Keywords: Local times,
Hilbert transform,
Ito formulaNature: Original Retrieve article from Numdam
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
2926Keywords: Brownian motion,
Several parameter processesNature: Original Retrieve article from Numdam
XVII: 22, 198-204, LNM 986 (1983)
KARANDIKAR, Rajeeva L.
Girsanov type formula for a Lie group valued Brownian motion (
Brownian motion,
Stochastic differential geometry)
A formula for the change of measure of a Lie group valued Brownian motion is stated and proved. It needs a Borel correspondence between paths in the Lie algebra and paths in the group, that transforms all (continuous) semimartingales in the algebra into their stochastic exponential
Comment: For more on stochastic exponentials in Lie groups, see Hakim-Dowek-Lépingle
2023 and Arnaudon
2612Keywords: Changes of measure,
Brownian motion in a manifold,
Lie groupNature: Original Retrieve article from Numdam
XVII: 24, 221-224, LNM 986 (1983)
BASS, Richard F.
Skorohod imbedding via stochastic integrals (
Brownian motion)
A centered probability $\mu$ on $\bf R$ is the law of $g(X_1)$, for a suitable function $g$ and $(X_t,\ t\le 1)$ a Brownian motion. The martingale with terminal value $g(X_1)$ is a time change $(T(t), \ t\le1)$ of a Brownian motion $\beta$; it is shown that $T(1)$ is a stopping time for $\beta$, thus showing the Skorohod embedding for $\mu$
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XVII: 25, 225-226, LNM 986 (1983)
MEILIJSON, Isaac
On the Azéma-Yor stopping time (
Brownian motion)
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XVII: 26, 227-239, LNM 986 (1983)
VALLOIS, Pierre
Le problème de Skorokhod sur $\bf R$ : une approche avec le temps local (
Brownian motion)
A solution to Skorohod's embedding problem is given, using the first hiting time of a set by the 2-dimensional process which consists of Brownian motion and its local time at zero. The author's aim is to ``correct'' the asymmetry inherent to the Azéma-Yor construction
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbedding,
Local timesNature: Original Retrieve article from Numdam
XIX: 27, 297-313, LNM 1123 (1985)
LE GALL, Jean-François
Sur la mesure de Hausdorff de la courbe brownienne (
Brownian motion)
Previous results on the $h$-measure of the Brownian curve in $
R^2$ or $
R^3$ indexed by $t\in[0,1]$, by Cisielski-Taylor
Trans. Amer. Math. Soc. 103 (1962) and Taylor
Proc. Cambridge Philos. Soc. 60 (1964) are sharpened. The method uses the description à la Ray-Knight of the local times of Bessel processes
Comment: These Ray-Knight descriptions are useful ; they were later used in questions not related to Hausdorff measures. See for instance Biane-Yor,
Ann. I.H.P. 23 (1987), Yor,
Ann. I.H.P. 27 (1991)
Keywords: Hausdorff measures,
Brownian motion,
Bessel processes,
Ray-Knight theoremsNature: Original Retrieve article from Numdam
XIX: 28, 314-331, LNM 1123 (1985)
LE GALL, Jean-François
Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan (
Brownian motion)
The normalized self-intersection local time of planar Brownian motion was shown to exist by Varadhan (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory, Academic Press, 1969). This is established anew here by a completely different method, using the intersection local time of two independent planar Brownian motions (whose existence was established by Geman, Horowitz and Rosen,
Ann. Prob. 12, 1984) and a sequence of dyadic decompositions of the triangle $\{0<s<t\le1\}$
Comment: Later, Dynkin, Rosen, Le Gall and others have shown existence of a renormalized local time for the multiple self-intersection of arbitrary order $n$ of planar Brownian motion. A good reference is Le Gall,
École d'Été de Saint-Flour XX, Springer LNM 1527
Keywords: Brownian motion,
Local times,
Self-intersectionNature: Original proofs Retrieve article from Numdam
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
1928Comment: 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-intersectionNature: Original proofs Retrieve article from Numdam
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-intersectionNature: Original Retrieve article from Numdam
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 $({\cal E}^x,\ x\in
R)$, 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 $(\tilde{\cal E}^x,\ x\in
R)$ 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,
FiltrationsNature: Original Retrieve article from Numdam
XX: 33, 515-531, LNM 1204 (1986)
ROSEN, Jay S.
A renormalized local time for multiple intersections of planar Brownian motion (
Brownian motion)
Using Fourier techniques, the existence of a renormalized local time for $n$-fold self-intersections of planar Brownian motion is obtained, thus extending the case $n=2$, obtained in the pioneering work of Varadhan (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory, Academic Press, 1969)
Comment: Closely related to
2036. A general reference is Le Gall,
École d'Été de Saint-Flour XX, Springer LNM 1527
Keywords: Local times,
Self-intersectionNature: Original Retrieve article from Numdam
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 representationNature: Original proofs Retrieve article from Numdam
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
3320Keywords: Maximal process,
Seshadri's identityNature: Original Retrieve article from Numdam
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
3325Keywords: Filtrations,
Spider martingales,
Walsh's Brownian motion,
Cosiness,
Slutsky's lemmaNature: New exposition of known results,
Original additions Retrieve article from Numdam
XXXIII: 04, 217-220, LNM 1709 (1999)
DE MEYER, Bernard
Une simplification de l'argument de Tsirelson sur le caractère non-brownien des processus de Walsh (
Brownian motion,
Filtrations)
Barlow's conjecture is proved with a simpler argument than in
3219Keywords: Filtrations,
Spider martingalesNature: New proof of known results Retrieve article from Numdam
XXXIII: 20, 388-394, LNM 1709 (1999)
PITMAN, James W.
The distribution of local times of a Brownian bridge (
Brownian motion)
Several useful identities for the one-dimensional marginals of local times of Brownian bridges are derived. This is a variation and extension on the well-known joint law of the maximum and the value of Brownian motion at a given time
Comment: Useful references are Borodin,
Russian Math. Surveys (1989) and the book
Brownian motion and stochastic calculus by Karatzas-Shrieve (Springer, 1991)
Keywords: Local times,
Brownian bridgeNature: Original Retrieve article from Numdam
XLI: 07, 161-179, LNM 1934 (2008)
BARAKA, Driss;
MOUNTFORD, Thomas
A law of the iterated logarithm for fractional Brownian motions (
Theory of fractional Brownian motion)
Nature: Original
XLI: 08, 181-197, LNM 1934 (2008)
NOURDIN, Ivan
A simple theory for the study of SDEs driven by a fractional Brownian motion in dimension one (
Theory of fractional Brownian motion)
Nature: Original
XLII: 14, 383-396, LNM 1978 (2009)
ÉMERY, Michel
Recognising whether a filtration is Brownian: a case study (
Theory of Brownian motion)
Keywords: Brownian filtrationNature: Original
XLIII: 03, 95-104, LNM 2006 (2011)
ROSEN, Jay
A stochastic calculus proof of the CLT for the $L^{2}$ modulus of continuity of local time (
Theory of Brownian motion)
Keywords: Central Limit Theorem,
Moduli of continuity,
Local times,
Brownian motionNature: Original
XLIII: 10, 241-268, LNM 2006 (2011)
BÉRARD BERGERY, Blandine;
VALLOIS, Pierre
Convergence at first and second order of some approximations of stochastic integrals (
Theory of Brownian motion,
Theory of stochastic integrals)
Keywords: Stochastic integration by regularization,
Quadratic variation,
First and second order convergence,
Stochastic Fubini's theoremNature: 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
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 transformNature: Original
