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7 matches found
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 processes
Nature: Original
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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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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 processes
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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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 2917
Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators
Nature: Original
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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 2916
Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators
Nature: Original
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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 Lévy processes)
Keywords: Stochastic integration, Lévy processes, Generalized Ornstein-Uhlenbeck processes
Nature: Original