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XVI: 01, 1-7, LNM 920 (1982)
TALAGRAND, Michel
Sur les résultats de Feyel concernant les épaisseurs
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XVI: 02, 8-28, LNM 920 (1982)
DELLACHERIE, Claude; FEYEL, Denis; MOKOBODZKI, Gabriel
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XVI: 03, 29-40, LNM 920 (1982)
DELLACHERIE, Claude
Appendice à l'exposé précédent
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XVI: 04, 41-90, LNM 920 (1982)
LONDON, R.R.; McKEAN, Henry P.; ROGERS, L.C.G.; WILLIAMS, David
A martingale approach to some Wiener-Hopf problems (two parts)
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XVI: 05, 91-94, LNM 920 (1982)
WILLIAMS, David
A potential-theoretic note on the quadratic Wiener-Hopf equation for Q-matrices
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XVI: 06, 95-132, LNM 920 (1982)
MEYER, Paul-André
Note sur les processus d'Ornstein-Uhlenbeck (Malliavin's calculus)
With every Gaussian measure $\mu$ one can associate an Ornstein-Uhlenbeck semigroup, for which $\mu$ is a reversible invariant measure. When $\mu$ is Wiener's measure on ${\cal C}(R)$, this semigroup is a fundamental tool in Malliavin's own approach to the Malliavin calculus''. See for instance Stroock's exposition of it in Math. Systems Theory, 13, 1981. With this semigroup one can associate its generator $L$ which plays the role of the classical Laplacian, and the positive bilinear functional $\Gamma(f,g)= L(fg)-fLg-gLf$---leaving aside domain problems for simplicity---sometimes called carré du champ'', which plays the role of the squared classical gradient. As in classical analysis, one can define it as $\sum_i \nabla_i f\nabla i g$, the derivatives being relative to an orthonormal basis of the Cameron-Martin space. We may define Sobolev-like spaces of order one in two ways: either by the fact that $Cf$ belongs to $L^p$, where $C=-\sqrt{-L}$ is the Cauchy generator'', or by the fact that $\sqrt{\Gamma(f,f)}$ belongs to $L^p$. A result which greatly simplifies the analytical part of the Malliavin calculus'' is the fact that both definitions are equivalent. This is the main topic of the paper, and its proof uses the Littlewood-Paley-Stein theory for semigroups as presented in 1010, 1510
Comment: An important problem is the extension to higher order Sobolev-like spaces. For instance, we could define the Sobolev space of order 2 either by the fact that $C^2f=-Lf$ belongs to $L^p$, and on the other hand define $\Gamma_2(f,g)=\sum_{ij} \nabla_i\nabla_j f \nabla_i\nabla_j g$ (derivatives of order 2) and ask that $\sqrt{\Gamma_2(f,f)}\in L^p$. For the equivalence of these two definitions and general higher order ones, see 1816, which anyhow contains many improvements over 1606. Also, proofs of these results have been given which do not involve Littlewood-Paley methods. For instance, Pisier has a proof which only uses the boundedness in $L^p$ of classical Riesz transforms.\par Another trend of research has been the correct definition of higher gradients'' within semigroup theory (the preceding definition of $\Gamma_2(f,g)$ makes use of the Gaussian structure). Bakry investigated the fundamental role of true'' $\Gamma_2$, the bilinear form $\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(Lf,g)$, which is positive in the case of the Ornstein-Uhlenbeck semigroup but is not always so. See 1909, 1910, 1912
Keywords: Ornstein-Uhlenbeck process, Gaussian measures, Littlewood-Paley theory, Hypercontractivity, Hermite polynomials, Riesz transforms, Test functions
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XVI: 07, 133-133, LNM 920 (1982)
MEYER, Paul-André
Appendice : Un résultat de D. Williams (Malliavin's calculus)
This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process
Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis
Nature: Exposition
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XVI: 08, 134-137, LNM 920 (1982)
BAKRY, Dominique
Remarques sur le processus d'Ornstein-Uhlenbeck en dimension infinie (Malliavin's calculus, Several parameter processes)
A process taking values in a space of sample paths can be considered as a two parameter process. Considering in this way the Ornstein-Uhlenbeck process (1606) raises a few natural questions, like the commutation of conditional expectations relative to the two filtrations---which is shown to hold true
Keywords: Ornstein-Uhlenbeck process
Nature: Original
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XVI: 09, 138-150, LNM 920 (1982)
BAKRY, Dominique; MEYER, Paul-André
Sur les inégalités de Sobolev logarithmiques (two parts) (Applications of martingale theory)
These two papers are variations on a paper of G.F. Feissner (Trans. Amer Math. Soc., 210, 1965). Let $\mu$ be a Gaussian measure, $P_t$ be the corresponding Ornstein-Uhlenbeck semigroup. Nelson's hypercontractivity theorem states (roughly) that $P_t$ is bounded from $L^p(\mu)$ to some $L^q(\mu)$ with $q\ge p$. In another celebrated paper, Gross showed this to be equivalent to a logarithmic Sobolev inequality, meaning that if a function $f$ is in $L^2$ as well as $Af$, where $A$ is the Ornstein-Uhlenbeck generator, then $f$ belongs to the Orlicz space $L^2Log_+L$. The starting point of Feissner was to translate this again as a result on the Riesz potentials'' of the semi-group (defined whenever $f\in L^2$ has integral $0$) $$R^{\alpha}={1\over \Gamma(\alpha)}\int_0^\infty t^{\alpha-1}P_t\,dt\;.$$ Note that $R^{\alpha}R^{\beta}=R^{\alpha+\beta}$. Then the theorem of Gross implies that $R^{1/2}$ is bounded from $L^2$ to $L^2Log_+L$. This suggests the following question: which are in general the smoothing properties of $R^\alpha$? (Feissner in fact considers a slightly different family of potentials).\par The complete result then is the following : for $\alpha$ complex, with real part $\ge0$, $R^\alpha$ is bounded from $L^pLog^r_+L$ to $L^pLog^{r+p\alpha}_+L$. The method uses complex interpolation between two cases: a generalization to Orlicz spaces of a result of Stein, when $\alpha$ is purely imaginary, and the case already known where $\alpha$ has real part $1/2$. The first of these two results, proved by martingale theory, is of a quite general nature
Keywords: Logarithmic Sobolev inequalities, Hypercontractivity, Gaussian measures, Riesz potentials
Nature: Original
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XVI: 10, 151-152, LNM 920 (1982)
MEYER, Paul-André
Sur une inégalité de Stein (Applications of martingale theory)
In his book Topics in harmonic analysis related to the Littlewood-Paley theory (1970) Stein uses interpolation between two results, one of which is a discrete martingale inequality deduced from the Burkholder inequalities, whose precise statement we omit. This note states and proves directly the continuous time analogue of this inequality---a mere exercise in translation
Keywords: Littlewood-Paley theory, Martingale inequalities
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XVI: 11, 153-158, LNM 920 (1982)
MEYER, Paul-André
Interpolation entre espaces d'Orlicz (Functional analysis)
This is an exposition of Calderon's complex interpolation method, in the case of moderate Orlicz spaces, aiming at its application in 1609
Keywords: Interpolation, Orlicz spaces, Moderate convex functions
Nature: Exposition
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XVI: 12, 159-183, LNM 920 (1982)
BRANCOVAN, Mihaï; BRONNER, François; PRIOURET, Pierre
Grandes déviations pour certains systèmes différentiels aléatoires
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XVI: 13, 184-200, LNM 920 (1982)
PRIOURET, Pierre
Remarques sur les petites perturbations de systèmes dynamiques
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XVI: 14, 201-208, LNM 920 (1982)
PERKINS, Edwin A.
Local time and pathwise uniqueness for stochastic differential equations
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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, Semimartingales
Nature: Original
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XVI: 16, 212-212, LNM 920 (1982)
WALSH, John B.
A non-reversible semi-martingale (Stochastic calculus)
A simple example is given of a continuous semimartingale (a Brownian motion which stops at time $1$ and starts moving again at time $T>1$, $T$ encoding all the information up to time $1$) whose reversed process is not a semimartingale
Keywords: Semimartingales, Time reversal
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: 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: 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: 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, 1321
Keywords: Multiplicative decomposition, Change of variable formula, Local times
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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XVI: 22, 248-256, LNM 920 (1982)
JEULIN, Thierry
Sur la convergence absolue de certaines intégrales (General theory of processes)
This paper is devoted to the a.s. absolute convergence of certain random integrals, a classical example of which is $\int_0^t ds/|B_s|^{\alpha}$ for Brownian motion starting from $0$. The author does not claim to prove deep results, but his technique of optional increasing reordering (réarrangement) of a process should be useful in other contexts too
Comment: This paper greatly simplifies a proof in the author's Semimartingales et Grossissement de Filtrations, LNM 833, p.44
Keywords: Enlargement of filtrations
Nature: Original
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XVI: 23, 257-267, LNM 920 (1982)
FLIESS, Michel; NORMAND-CYROT, Dorothée
Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T.~Chen (Stochastic calculus)
Consider a s.d.e. in a manifold, $dX_t=\sum_i A_i(X)\,dM^i_t$ (Stratonovich differentials), driven by continuous real semimartingales $M^i_t$, and where the $A_i$ have the geometrical nature of vector fields. Such an equation has a counterpart in which the $M^i(t)$ are arbitrary deterministic piecewise smooth functions, and if this equation can be solved by some deterministic machinery, then the s.d.e. can be solved too, just by making the input random. Thus a bridge is drawn between s.d.e.'s and problems of deterministic control theory. From this point of view, the complexity of the problem reflects that of the Lie algebra generated by the vector fields $A_i$. Assuming these fields are complete (i.e., generate true one-parameter groups on the manifold) and generate a finite dimensional Lie algebra (which then is the Lie algebra of a matrix group), the problem can be linearized. If the Lie algebra is nilpotent, the solution then can be expressed explicitly as a function of a finite number of iterated integrals of the driving processes (Chen integrals), and this provides the required deterministic machine''. It thus appears that results like those of Yamato (Zeit. für W-Theorie, 47, 1979) do not really belong to probability theory
Comment: See Kunita's paper 1432
Keywords: Stochastic differential equations, Lie algebras, Chen's iterated integrals, Campbell-Hausdorff formula
Nature: Original
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XVI: 24, 268-284, LNM 920 (1982)
UPPMAN, Are
Sur le flot d'une équation différentielle stochastique (Stochastic calculus)
This paper is a companion to 1506, devoted to the main results on the flow of a (Lipschitz) stochastic differential equation driven by continous semimartingales: non-confluence of solutions from different initial points, surjectivity of the mapping, smooth dependence on the initial conditions. The proofs have been greatly simplified
Keywords: Stochastic differential equations, Flow of a s.d.e., Injectivity
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XVI: 25, 285-297, LNM 920 (1982)
UPPMAN, Are
Un théorème de Helly pour les surmartingales fortes (Martingale theory)
Provide the set of (optional) strong supermartingales $X$ of the class (D) with the topology of weak $L^1$--convergence of $X_T$ at each stopping time $T$. Then it is shown that any subset which belongs uniformly to the class (D) is relatively compact, also in the sequential sense of extracting convergent subsequences
Comment: This paper was suggested by a similar result of Mokobodzki for strongly supermedian functions in potential theory
Keywords: Supermartingales, Strong supermartingales
Nature: Original
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XVI: 26, 298-313, LNM 920 (1982)
DELLACHERIE, Claude; LENGLART, Érik
Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (General theory of processes)
This paper is a sequel to 1524. Let $\Theta$ be a chronology, i.e., a family of stopping times containing $0$ and $\infty$ and closed under the operations $\land,\lor$---examples are the family of all stopping times, and that of all deterministic stopping times. The general problem discussed is that of defining an optional process $X$ on $[0,\infty]$ such that for each $T\in\Theta$ $X_T$ is a.s. equal to a given r.v. (${\cal F}_T$-measurable). While in 1525 the discussion concerned supermartingales, it is extended here to processes which satisfy a semi-continuity condition from the right
Keywords: Stopping times
Nature: Original
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XVI: 27, 314-318, LNM 920 (1982)
LENGLART, Érik
Sur le théorème de la convergence dominée (General theory of processes, Stochastic calculus)
Consider previsible processes $U^n,U$ such that $U^n_T\rightarrow U_T$ in some sense at bounded previsible times $T$. The problem discussed is whether stochastic integrals $\int U^n_s dX_s$ converge (in the same sense) to $\int U_sdX_s$. Under a domination hypothesis, the answer is shown to be positive if the convergence is either weak convergence in $L^1$, or convergence in probability. The existence of the limiting process $U$ is not assumed in the paper; it is proved by a modification of an argument of Mokobodzki for which see 1110
Keywords: Stopping times, Optional processes, Weak convergence, Stochastic integrals
Nature: Original
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XVI: 28, 319-337, LNM 920 (1982)
EAGLESON, G.K.; MÉMIN, Jean
Sur la contiguïté de deux suites de mesures : généralisation d'un théorème de Kabanov-Liptser-Shiryayev
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XVI: 29, 338-347, LNM 920 (1982)
YAN, Jia-An
À propos de l'intégrabilité uniforme des martingales exponentielles (Martingale theory)
Sufficient conditions are given for the uniform integrability of the exponential ${\cal E}(M)$, where $M$ is a local martingale with jumps $\ge-1$, refining older results of Lépingle and Mémin, and of the author. They involve the Lévy measure of the martingale
Comment: In the lemma p.339 delete the assumption $0<\beta$
Keywords: Exponential martingales
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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XVI: 31, 355-369, LNM 920 (1982)
BAKRY, Dominique
Semimartingales à deux indices
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XVI: 32, 370-379, LNM 920 (1982)
ZHENG, Wei-An
Semimartingales in predictable random open sets
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XVI: 33, 380-383, LNM 920 (1982)
ABOULAÏCH, Rajae
Intégrales stochastiques généralisées
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XVI: 34, 384-391, LNM 920 (1982)
KARANDIKAR, Rajeeva L.
A.s. approximation results for multiplicative stochastic integrals
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XVI: 35, 392-399, LNM 920 (1982)
MEILIJSON, Isaac
There exists no ultimate solution to Skorokhod's problem
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XVI: 36, 400-408, LNM 920 (1982)
EL KAROUI, Nicole
Une propriété de domination de l'enveloppe de Snell des semimartingales fortes
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XVI: 37, 409-411, LNM 920 (1982)
CHOU, Ching Sung
Une remarque sur l'approximation des solutions d'e.d.s.
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XVI: 38, 412-441, LNM 920 (1982)
On some limit theorems for solutions of stochastic differential equations
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XVI: 39, 442-446, LNM 920 (1982)
JACOD, Jean
Équations différentielles linéaires : la méthode de variation des constantes
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XVI: 40, 447-450, LNM 920 (1982)
JACOD, Jean; PROTTER, Philip
Quelques remarques sur un nouveau type d'équations différentielles stochastiques
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XVI: 41, 451-468, LNM 920 (1982)
PROTTER, Philip
Stochastic differential equations with feedback in the differentials
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XVI: 42, 469-489, LNM 920 (1982)
PELLAUMAIL, Jean
Règle maximale
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XVI: 43, 490-502, LNM 920 (1982)
MÉTIVIER, Michel
Pathwise differentiability with respect to a parameter of solutions of stochastic differential equations
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XVI: 44, 503-508, LNM 920 (1982)
MEYER, Paul-André
Résultats d'Atkinson sur les processus de Markov
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XVI: 45, 509-514, LNM 920 (1982)
GRAVERSEN, Svend Erik; RAO, Murali
Hypothesis (B) of Hunt
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XVI: 46, 515-518, LNM 920 (1982)
GLOVER, Joseph
An extension of Motoo's theorem
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XVI: 47, 519-543, LNM 920 (1982)
GHOUSSOUB, Nassif
An integral representation of randomized probabilities and its applications
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XVI: 48, 544-569, LNM 920 (1982)
CHEVET, Simone
Topologies métrisables rendant continues les trajectoires d'un processus
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XVI: 49, 570-580, LNM 920 (1982)
CHATTERJI, Shrishti Dhav; RAMASWAMY, S.
Mesures gaussiennes et mesures produits
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XVI: 50, 581-601, LNM 920 (1982)
EHRHARD, Antoine
Sur la densité du maximum d'une fonction aléatoire gaussienne
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XVI: 51, 602-608, LNM 920 (1982)
HEINKEL, Bernard
Sur la loi du logarithme itéré dans les espaces réflexifs
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XVI: 52, 609-622, LNM 920 (1982)
LEDOUX, Michel
La loi du logarithme itéré pour les variables aléatoires prégaussiennes à valeurs dans un espace de Banach à norme régulière
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XVI: 53, 623-623, LNM 920 (1982)
MEYER, Paul-André
Correction au Séminaire XV
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XVI: 54, 623-623, LNM 920 (1982)
MEYER, Paul-André