XIV: 01, 1-16, LNM 784 (1980)
HEINKEL, Bernard
Deux exemples d'utilisation de mesures majorantes (
Gaussian processes)
From the introduction: ``our purpose is to study in detail two examples to which the method of majorizing measures, but not the entropy method, can be applied''
Nature: Original
Retrieve article from Numdam
XIV: 02, 17-17, LNM 784 (1980)
GINÉ, Evarist
Corrections to ``Domains of attraction in Banach spaces'' (
Gaussian processes)
Contains three minor corrections to
1303Nature: Correction Retrieve article from Numdam
XIV: 03, 18-25, LNM 784 (1980)
CAIROLI, Renzo
Sur l'extension de la définition d'intégrale stochastique (
Several parameter processes)
A result of Wong-Zakai (
Ann. Prob. 5, 1977) extending the definition of the two kinds of stochastic integrals relative to the Brownian sheet is generalized to cover the case of stochastic integration relative to martingales, or strong martingales
Comment: A note at the end of the paper suggests some improvements
Keywords: Stochastic integrals,
Brownian sheetNature: Original Retrieve article from Numdam
XIV: 04, 26-48, LNM 784 (1980)
LENGLART, Érik;
LÉPINGLE, Dominique;
PRATELLI, Maurizio
Présentation unifiée de certaines inégalités de la théorie des martingales (
Martingale theory)
This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes
Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,
Probabilités et Potentiel E ; see also
1621. A striking example of the power of these methods is Barlow-Yor, {\sl Jour. Funct. Anal.}
49,1982
Keywords: Moderate convex functions,
Inequalities,
Martingale inequalities,
Burkholder inequalities,
Good lambda inequalities,
Domination inequalitiesNature: Original Retrieve article from Numdam
XIV: 05, 49-52, LNM 784 (1980)
LENGLART, Érik
Appendice à l'exposé précédent~: inégalités de semimartingales (
Martingale theory,
Stochastic calculus)
This paper contains several applications of the methods of
1404 to the case of semimartingales instead of martingales
Keywords: Inequalities,
SemimartingalesNature: Original Retrieve article from Numdam
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
1404Comment: Generalized by Takaoka
3313Keywords: Exponential martingales,
Continuous martingalesNature: Original Retrieve article from Numdam
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 theoremNature: Original Retrieve article from Numdam
XIV: 08, 76-101, LNM 784 (1980)
SHARPE, Michael J.
Local times and singularities of continuous local martingales (
Martingale theory)
This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh
1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$
Comment: See Calais-Genin
1717Keywords: Local times,
Local martingales,
Semimartingales in an open intervalNature: Original Retrieve article from Numdam
XIV: 09, 102-103, LNM 784 (1980)
MEYER, Paul-André
Sur un résultat de L. Schwartz (
Martingale theory)
the following definition of a semimartingale $X$ in a random open set $A$ is due to L. Schwartz (
Semimartingales dans les variétés..., Lecture Notes in M.
780): $A$ can be represented as a countable union of random open sets $A_n$, and for each $n$ there exists an ordinary semimartingale $Y_n$ such $X=Y_n$ on $A_n$. It is shown that if $K\subset A$ is a compact optional set, then there exists an ordinary semimartingale $Y$ such that $X=Y$ on $K$
Comment: The results are extended in Meyer-Stricker
Stochastic Analysis and Applications, part B, Advances in M. Supplementary Studies, 1981
Keywords: Semimartingales in a random open setNature: Exposition,
Original additions Retrieve article from Numdam
XIV: 10, 104-111, LNM 784 (1980)
STRICKER, Christophe
Prolongement des semi-martingales (
Stochastic calculus)
The problem consists in characterizing semimartingales on $]0,\infty[$ which can be ``closed at infinity'', and the similar problem at $0$. The criteria are similar to the Vitali-Hahn-Saks theorem and involve convergence in probability of suitable stochastic integrals. The proof rests on a functional analytic result of Maurey-Pisier
Keywords: Semimartingales,
Semimartingales in an open intervalNature: Original Retrieve article from Numdam
XIV: 11, 112-115, LNM 784 (1980)
STRICKER, Christophe
Projection optionnelle des semi-martingales (
Stochastic calculus)
Let $({\cal G}_t)$ be a subfiltration of $({\cal F}_t)$. Since the optional projection on $({\cal G}_t)$ of a ${\cal F}$-martingale is a ${\cal G}$-martingale, and the projection of an increasing process a ${\cal G}$-submartingale, projections of ${\cal F}$-semimartingales ``should be'' ${\cal G}$-semimartingales. This is true for quasimartingales, but false in general
Comment: The main results on subfiltrations are proved by Stricker in
Zeit. für W-Theorie, 39, 1977
Keywords: Semimartingales,
Projection theoremsNature: Original Retrieve article from Numdam
XIV: 12, 116-117, LNM 784 (1980)
CHOU, Ching Sung
Une caractérisation des semimartingales spéciales (
Stochastic calculus)
This is a useful addition to the next paper
1413: a semimartingale can be ``controlled'' (in the sense of Métivier-Pellaumail) by a locally integrable increasing process if and only if it is special
Comment: See also
1352Keywords: Semimartingales,
Métivier-Pellaumail inequality,
Special semimartingalesNature: Original Retrieve article from Numdam
XIV: 13, 118-124, LNM 784 (1980)
ÉMERY, Michel
Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (
Stochastic calculus)
Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see
1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales
Comment: See
1352. A general reference on the Métivier-Pellaumail method can be found in their book
Stochastic Integration, Academic Press 1980. See also He-Wang-Yan,
Semimartingale Theory and Stochastic Calculus, CRC Press 1992
Keywords: Semimartingales,
Spaces of semimartingales,
Stochastic differential equations,
Doob's inequality,
Métivier-Pellaumail inequalityNature: Original Retrieve article from Numdam
XIV: 14, 125-127, LNM 784 (1980)
LENGLART, Érik
Sur l'inégalité de Métivier-Pellaumail (
Stochastic calculus)
A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality
Keywords: Doob's inequality,
Métivier-Pellaumail inequalityNature: New proof of known results Retrieve article from Numdam
XIV: 15, 128-139, LNM 784 (1980)
CHOU, Ching Sung;
MEYER, Paul-André;
STRICKER, Christophe
Sur l'intégrale stochastique de processus prévisibles non bornés (
Stochastic calculus)
The standard theory of stochastic integration deals with locally bounded previsible processes. The natural definition of the stochastic integral $H.X$ of a previsible process $H$ w.r.t. a semimartingale $X$ consists in assuming the existence of some decomposition $X=M+A$ such that $H.M$ exists in the martingale sense, and $H.A$ in the Stieltjes sense, and then defining $H.X$ as their sum. This turns out to be a very awkward definition. It is shown here to be equivalent to the following one: truncating $H$ at $n$, the standard stochastic integrals $H_n.X$ converge in the topology of semimartingales. This is clearly invariant under changes of law. A counterexample shows that integrability may be lost if the filtration is enlarged
Comment: See also
1417. This is a synthesis of earlier work, much of which is due to Jacod,
Calcul Stochastique et Problèmes de Martingales, Lect. Notes in M. 714. The contents of this paper appeared in book form in Dellacherie-Meyer,
Probabilités et Potentiel B, Chap. VIII, \S3. An equivalent definition is given by L. Schwartz in
1530, using the idea of ``formal semimartingales''. For further steps in the same direction, see Stricker
1533Keywords: Stochastic integralsNature: Exposition,
Original additions Retrieve article from Numdam
XIV: 16, 140-147, LNM 784 (1980)
ÉMERY, Michel
Métrisabilité de quelques espaces de processus aléatoires (
General theory of processes,
Stochastic calculus)
As a sequel to the main work of
1324 on the topology of semimartingales, several spaces of processes defined by localization (or prelocalization) of standard spaces of martingales or processes of bounded variation are studied here, and shown to be metrizable and complete
Keywords: Spaces of semimartingalesNature: Original Retrieve article from Numdam
XIV: 17, 148-151, LNM 784 (1980)
YAN, Jia-An
Remarques sur l'intégrale stochastique de processus non bornés (
Stochastic calculus)
It is shown how to develop the integration theory of unbounded previsible processes (due to Jacod
1126), starting from the elementary definition considered ``awkward'' in
1415Comment: Another approach to those integrals is due to L. Schwartz, in his article
1530 on formal semimartingales
Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XIV: 18, 152-160, LNM 784 (1980)
ÉMERY, Michel
Compensation de processus à variation finie non localement intégrables (
General theory of processes,
Stochastic calculus)
First an example is given of a local martingale $M$ and an unbounded previsible process $H$ such that $H.M$ exists in the sense of
1126 and
1415, but is not a local martingale. This leads to a natural enlargement of the class of local martingales, which turns out to be the same suggested by Chou in
1322 under the name of class $(\Sigma_m)$. Once the class has been so extended, the operation of previsible compensation can be extended to a class of processes with finite variation, but not locally integrable variation, and the class of special semimartingales can be also enlarged
Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997). Using the language of L. Schwartz
1530, it is the intersection of the set of (usual) semimartingales with the set of formal martingales
Keywords: Local martingales,
Stochastic integrals,
CompensatorsNature: Original Retrieve article from Numdam
XIV: 19, 161-172, LNM 784 (1980)
JACOD, Jean
Intégrales stochastiques par rapport à une semi-martingale vectorielle et changements de filtration (
Stochastic calculus,
General theory of processes)
Given a square integrable vector martingale $M$ and a previsible vector process $H$, the conditions implying the existence of the (scalar valued) stochastic integral $H.M$ are less restrictive than the existence of the ``componentwise'' stochastic integral, unless the components of $M$ are orthogonal (this result was due to Galtchouk, 1975). The theory of vector stochastic integrals, though parallel to the scalar theory, requires a careful theory given in this paper
Comment: Another approach, yielding an equivalent definition, is followed by L. Schwartz in his article
1530 on formal semimartingales
Keywords: Semimartingales,
Stochastic integralsNature: Original Retrieve article from Numdam
XIV: 20, 173-188, LNM 784 (1980)
MEYER, Paul-André
Les résultats de Jeulin sur le grossissement des tribus (
General theory of processes,
Stochastic calculus)
This is an introduction to beautiful results of Jeulin on enlargements, for which see
Zeit. für W-Theorie, 52, 1980, and above all the Lecture Notes vol. 833,
Semimartingales et grossissement d'une filtration Comment: See also
1329,
1350Keywords: Enlargement of filtrations,
SemimartingalesNature: Exposition 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: 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 Retrieve article from Numdam
XIV: 23, 205-208, LNM 784 (1980)
SEYNOU, Aboubakary
Sur la compatibilité temporelle d'une tribu et d'une filtration discrète (
General theory of processes)
Let us say that a $\sigma$-field ${\cal G}$ is embedded into a filtration $({\cal H}_n)$ if ${\cal G}= {\cal H}_T$ for some ${\cal H}$-stopping time $T$, that a filtration $({\cal F}_n)$ is embedded into $({\cal H}_n)$ if ${\cal F}_S$ can be embedded into $({\cal H}_n)$ for every ${\cal F}$-stopping time $S$, and finally that ${\cal G}$ and ${\cal F}$ or $({\cal F}_n)$ are compatible if they can be embedded into one single filtration $({\cal H}_n)$. Then the question investigated is whether the compatibility of ${\cal G}$ and the individual $\sigma$-fields ${\cal F}_n$ implies that of ${\cal G}$ and the whole filtration $({\cal F}_n)$
Comment: This problem arose from the spectral point of view on stochastic integration as in
1123Keywords: FiltrationsNature: Original Retrieve article from Numdam
XIV: 24, 209-219, LNM 784 (1980)
PELLAUMAIL, Jean
Remarques sur l'intégrale stochastique (
Stochastic calculus)
This is an exposition of stochastic integrals and stochastic differential equations for Banach space valued processes along the lines of Métivier-Pellaumail
Stochastic Integration (1980), the class of semimartingales being defined by the Métivier-Pellaumail inequality (
1413)
Keywords: Stochastic integrals,
Stochastic differential equations,
Métivier-Pellaumail inequalityNature: Exposition Retrieve article from Numdam
XIV: 25, 220-222, LNM 784 (1980)
YAN, Jia-An
Caractérisation d'ensembles convexes de $L^1$ ou $H^1$ (
Stochastic calculus,
Functional analysis)
This is a new and simpler approach to the crucial functional analytic lemma in
1354 (the proof that semimartingales are the stochastic integrators in $L^0$). That is, given a convex set $K\subset L^1$ containing $0$, find a condition for the existence of $Z>0$ in $L^\infty$ such that $\sup_{X\in K}E[ZX]<\infty$. A similar result is discussed for $H^1$ instead of $L^1$
Comment: This lemma, instead of the original one, has proved very useful in mathematical finance
Keywords: Semimartingales,
Stochastic integrals,
Convex functionsNature: Original Retrieve article from Numdam
XIV: 26, 223-226, LNM 784 (1980)
YAN, Jia-An
Remarques sur certaines classes de semimartingales et sur les intégrales stochastiques optionnelles (
Stochastic calculus)
A class of semimartingales containing the special ones is introduced, which can be intrinsically decomposed into a continuous and a purely discontinuous part. These semimartingales have ``not too large totally inaccessible jumps''. In the second part of the paper, a non-compensated optional stochastic integral is defined, improving the results of Yor
1335Keywords: Semimartingales,
Optional stochastic integralsNature: Original Retrieve article from Numdam
XIV: 27, 227-248, LNM 784 (1980)
JACOD, Jean;
MÉMIN, Jean
Sur la convergence des semimartingales vers un processus à accroissements indépendants (
General theory of processes,
Stochastic calculus,
Martingale theory)
A method of Kabanov, Liptzer and Shiryaev is adapted to study the convergence of a sequence of semimartingales to a process with independent increments (to be completed)
Keywords: Convergence in law,
TightnessNature: Original Retrieve article from Numdam
XIV: 28, 249-253, LNM 784 (1980)
YOEURP, Chantha
Sur la dérivation des intégrales stochastiques (
Stochastic calculus)
The following problem is discussed: under which conditions do ratios of the form $\int_t^{t+h} H_s\,dM_s/(M_{t+h}-M_t)$ converge to $H_t$ as $h\rightarrow 0$? It is shown that positive results due to Isaacson (
Ann. Math. Stat. 40, 1979) in the Brownian case fail in more general situations
Comment: See also
1529Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XIV: 29, 254-254, LNM 784 (1980)
YOEURP, Chantha
Rectificatif à l'exposé de C.S. Chou (
Stochastic calculus)
A mistake in the proof of
1337 is corrected, the result remaining true without additional assumptions
Keywords: Local times,
Semimartingales,
JumpsNature: Correction Retrieve article from Numdam
XIV: 30, 255-255, LNM 784 (1980)
REBOLLEDO, Rolando
Corrections à ``Décomposition des martingales locales et raréfaction des sauts'' (
General theory of processes,
Martingale theory)
Concerns
1311. For the definitive version, see
Mém. Soc. Math. France, 62, 1979
Keywords: Central limit theorem,
Skorohod topology,
Local martingales,
JumpsNature: Correction Retrieve article from Numdam
XIV: 31, 256-281, LNM 784 (1980)
FUJISAKI, Masatoshi
Contrôle stochastique continu et martingales (
General theory of processes)
To be completed
Keywords: Control theoryNature: Original Retrieve article from Numdam
XIV: 32, 282-304, LNM 784 (1980)
KUNITA, Hiroshi
On the representation of solutions of stochastic differential equations (
Stochastic calculus)
This paper concerns stochastic differential equations in the standard form $dY_t=\sum_i X_i(Y_t)\,dB^i(t)+X_0(Y_t)\,dt$ where the $B^i$ are independent Brownian motions, the stochastic integrals are in the Stratonovich sense, and $X_i,X_0$ have the geometric nature of vector fields. The problem is to find a deterministic (and smooth) machinery which, given the paths $B^i(.)$ will produce the path $Y(.)$. The complexity of this machinery reflects that of the Lie algebra generated by the vector fields. After a study of the commutative case, a paper of Yamato settled the case of a nilpotent Lie algebra, and the present paper deals with the solvable case. This line of thought led to the important and popular theory of flows of diffeomorphisms associated with a stochastic differential equation (see for instance Kunita's paper in
Stochastic Integrals, Lecture Notes in M. 851)
Comment: On a closely related subject, see the paper of Fliess and Norman-Cyrot,
1623Keywords: Stochastic differential equations,
Lie algebras,
Campbell-Hausdorff formulaNature: Original Retrieve article from Numdam
XIV: 33, 305-315, LNM 784 (1980)
YAN, Jia-An
Sur une équation différentielle stochastique générale (
Stochastic calculus)
The differential equation considered is of the form $X_t= \Phi(X)_t+\int_0^tF(X)_s\,dM_s$, where $M$ is a semimartingale, $\Phi$ maps adapted cadlag processes into themselves, and $F$ maps adapted cadlag process into previsible processes---not locally bounded, this is the main technical point. Some kind of Lipschitz condition being assumed, existence, uniqueness and stability are proved
Keywords: Stochastic differential equationsNature: Original Retrieve article from Numdam
XIV: 34, 316-317, LNM 784 (1980)
ÉMERY, Michel
Une propriété des temps prévisibles (
General theory of processes)
The idea is to prove that the general theory of processes on a random interval $[0,T[$, where $T$ is previsible, is essentially the same as on $[0,\infty[$. To this order, a continuous, strictly increasing adapted process $(A_t)$ is constructed, such that $A_0=0$, $A_T=1$
Keywords: Previsible timesNature: Original Retrieve article from Numdam
XIV: 35, 318-323, LNM 784 (1980)
ÉMERY, Michel
Annonçabilité des temps prévisibles. Deux contre-exemples (
General theory of processes)
It is shown that two standard results on previsible stopping times on a probability space, namely that every previsible time $T$ can be ``foretold'' by a strictly increasing sequence $T_n\uparrow T$, and that the $T_n$ can themselves be taken previsible, become false if exceptional sets of measure zero are not allowed
Keywords: Previsible timesNature: Original Retrieve article from Numdam
XIV: 36, 324-331, LNM 784 (1980)
BARLOW, Martin T.;
ROGERS, L.C.G.;
WILLIAMS, David
Wiener-Hopf factorization for matrices (
Markov processes)
Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix
Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in
1437Keywords: Wiener-Hopf factorizations,
Additive functionals,
Changes of time,
Markov chainsNature: Original Retrieve article from Numdam
XIV: 37, 332-342, LNM 784 (1980)
ROGERS, L.C.G.;
WILLIAMS, David
Time-substitution based on fluctuating additive functionals (Wiener-Hopf factorization for infinitesimal generators) (
Markov processes)
This is a first step towards the extension of
1436 to Markov processes with a general state space
Keywords: Wiener-Hopf factorizations,
Additive functionals,
Changes of timeNature: 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: 39, 347-356, LNM 784 (1980)
CHUNG, Kai Lai
On stopped Feynman-Kac functionals (
Markov processes,
Diffusion theory)
Let $(X_t)$ be a strong Markov process with continuous paths on the line, and let $\tau_b$ be the hitting time of the point $b$. It is assumed that $\tau_b$ is $P_a$-a.s. finite for all $a,b$. The purpose of the paper is to study the quantities $u(a,b)=E_a[\,\exp(\int_0^{\tau_b} q(X_s)\,ds)\,]$ where $q$ is bounded. Then (among other results) if $u(a,b)<\infty$ for all $a<b$, we have $u(a,b)\,u(b,a)\le 1$ for all $a,b$
Keywords: Hitting probabilitiesNature: 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
XIV: 42, 397-409, LNM 784 (1980)
GETOOR, Ronald K.
Transience and recurrence of Markov processes (
Markov processes)
From the introduction: The purpose of this paper is to present an elementary exposition of some various conditions that have been used to define transience or recurrence of a Markov process... an elementary and unified discussion of these ideas may be worthwhile
Keywords: Recurrent Markov processesNature: Exposition,
Original additions Retrieve article from Numdam
XIV: 43, 410-417, LNM 784 (1980)
JACOD, Jean;
MAISONNEUVE, Bernard
Remarque sur les fonctionnelles additives non adaptées des processus de Markov (
Markov processes)
It occurs sometimes that a Markov process $(X_t)$ satisfies in a filtration ${\cal H}_t$ a Markov property of the form $E[f\circ \theta_t \,|\,{\cal H}_t]= E_{X_t}[f]$, where $f$ is not restricted to be ${\cal H}_t$-measurable. For instance, situations in renewal theory where one is given a Markov pair $(X_t,Y_t)$, and ${\cal H}_t$ describes the path of $X$ up to time $t$, and the whole path of $Y$. In such cases, the authors show that additive functionals which are previsible in the larger filtration are in fact previsible in the filtration of $X$ alone
Keywords: Additive functionalsNature: Original Retrieve article from Numdam
XIV: 44, 418-436, LNM 784 (1980)
RAO, Murali
A note on Revuz measure (
Markov processes,
Potential theory)
The problem is to weaken the hypotheses of Chung (
Ann. Inst. Fourier, 23, 1973) implying the representation of the equilibrium potential of a compact set as a Green potential. To this order, Revuz measure techniques are used, and interesting auxiliary results are proved concerning the Revuz measures of natural additive functionals of a Hunt process
Keywords: Revuz measures,
Additive functionals,
Hunt processes,
Equilibrium potentialsNature: Original Retrieve article from Numdam
XIV: 45, 437-474, LNM 784 (1980)
TAKSAR, Michael I.
Regenerative sets on real line (
Markov processes,
Renewal theory)
From the introduction: A number of papers are devoted to studying regenerative sets on a positive half-line... our objective is to construct translation invariant sets of this type on the entire real line. Besides we start from a weaker definition of regenerativity
Comment: This important paper, if written in recent years, would have merged into the theory of Kuznetsov measures
Keywords: Regenerative setsNature: Original Retrieve article from Numdam
XIV: 46, 475-488, LNM 784 (1980)
WEBER, Michel
Sur un théorème de Maruyama (
Gaussian processes,
Ergodic theory)
Given a stationary centered Gaussian process $X$ with spectral measure $\mu$, a new proof is given of the fact that if $\mu$ is continuous, the flow of $X$ is weakly mixing
Keywords: Stationary processesNature: Original Retrieve article from Numdam
XIV: 47, 489-495, LNM 784 (1980)
CAIROLI, Renzo
Intégrale stochastique curviligne le long d'une courbe rectifiable (
Several parameter processes)
The problem is to define stochastic integrals $\int_{\partial A} \phi\,\partial_1W$ where $W$ is the Brownian sheet, $\phi$ is a suitable process, and $A$ a suitable domain of the plane with rectifiable boundary
Keywords: Stochastic integrals,
Brownian sheetNature: Original Retrieve article from Numdam
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 timeNature: Original Retrieve article from Numdam
XIV: 49, 500-546, LNM 784 (1980)
LENGLART, Érik
Tribus de Meyer et théorie des processus (
General theory of processes,
Stochastic calculus)
The subject of this paper is the study of the $\sigma$-field on $
R_+\times\Omega$ generated by a family of cadlag processes including the deterministic ones, and stable under stopping at non-random times. Of course the optional and previsible $\sigma$-fields are Meyer $\sigma$-fields in this very general sense. It is a matter of wonder to see how far one can go with such simple hypotheses, which were suggested by Dellacherie
705Comment: This beautiful paper was generally ignored. If a suggestive name had been used instead of the terminology ``Meyer $\sigma$-field'', its fate might have been different. See
1524 for an interesting application. The work of Fourati (partly unpublished) follows along the same lines, but including time reversal: see
2119Keywords: Projection theorems,
Section theoremsNature: Original Retrieve article from Numdam
