VIII: 07, 37-77, LNM 381 (1974)
DUPUIS, Claire
Mesure de Hausdorff de la trajectoire de certains processus à accroissements indépendants et stationnaires (
Independent increments)
The problem is to show that, for symmetric Lévy processes with small jumps and without a Brownian part, there exists a natural Hausdorff measure for which almost every path up to time $t$ has ``length'' exactly $t$. The case of Brownian motion had been known for a long time, the case of stable processes was settled by S.J.~Taylor (
J. Math. Mech.,
16, 1967) whose methods are generalized here
Keywords: Hausdorff measures,
Lévy processesNature: Original
Retrieve article from Numdam
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
1123Comment: 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
1221Keywords: Previsible representation,
Extreme points,
Independent increments,
Lévy processesNature: Original Retrieve article from Numdam
XIII: 10, 132-137, LNM 721 (1979)
SIDIBÉ, Ramatoulaye
Martingales locales à accroissements indépendants (
Martingale theory,
Independent increments)
It is shown here that a process with (stationary) independent increments which is a local martingale must be a true martingale
Comment: The case of non-stationary increments is considered in
1544. See also the errata sheet of vol. XV
Keywords: Local martingales,
Lévy processesNature: Original Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
XXIX: 16, 166-180, LNM 1613 (1995)
APPLEBAUM, David
A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (
Stochastic differential geometry,
Markov processes)
This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.
Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (
Stochastics Stochastics Rep. 56, 1996). The same question is addressed by Cohen in the next article
2917Keywords: Semimartingales with jumps,
Lévy processes,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
XXIX: 17, 181-193, LNM 1613 (1995)
COHEN, Serge
Some Markov properties of stochastic differential equations with jumps (
Stochastic differential geometry,
Markov processes)
The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see
1505 and
1655) was extended by Cohen to càdlàg semimartingales (
Stochastics Stochastics Rep. 56, 1996). Here this language is used to study the Markov property of solutions to SDE's with jumps. In particular,two definitions of a Lévy process in a Riemannian manifold are compared: One as the solution to a SDE driven by some Euclidean Lévy process, the other by subordinating some Riemannian Brownian motion. It is shown that in general the former is not of the second kind
Comment: The first definition is independently introduced by David Applebaum
2916Keywords: Semimartingales with jumps,
Lévy processes,
Subordination,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
XLIV: 03, 61-74, LNM 2046 (2012)
BASSE-O'CONNOR, Andreas;
GRAVERSEN, Svend-Erik;
PEDERSEN, Jan
Some classes of proper integrals and generalized Ornstein-Uhlenbeck processes (
Theory of processes)
Keywords: Stochastic integration,
Lévy processes,
Generalized Ornstein-Uhlenbeck processesNature: Original
XLV: 10, 277-300, LNM 2078 (2013)
DONEY, R. A.;
VAKEROUDIS, S.
Windings of Planar Stable Processes (
Theory of processes)
Keywords: Stable processes,
Lévy processes,
Brownian motion,
windings,
exit time from a cone,
Spitzer's Theorem,
skew-product representation,
Lamperti's relation,
Law of the Iterated Logarithm for small timesNature: Original
