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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 processes)

Keywords: Stochastic integration, Lévy processes, Generalized Ornstein-Uhlenbeck processes

Nature: 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 times

Nature: Original

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 (

Keywords: Hausdorff measures, Lévy processes

Nature: Original

Retrieve article from Numdam

XI: 35, 502-517, LNM 581 (1977)

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,

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

Retrieve article from Numdam

XIII: 10, 132-137, LNM 721 (1979)

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

Retrieve article from Numdam

XVI: 30, 348-354, LNM 920 (1982)

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

Retrieve article from Numdam

XXIX: 16, 166-180, LNM 1613 (1995)

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 (

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

Retrieve article from Numdam

XXIX: 17, 181-193, LNM 1613 (1995)

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 (

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

Retrieve article from Numdam

XLIV: 03, 61-74, LNM 2046 (2012)

Some classes of proper integrals and generalized Ornstein-Uhlenbeck processes (Theory of processes)

Keywords: Stochastic integration, Lévy processes, Generalized Ornstein-Uhlenbeck processes

Nature: Original

XLV: 10, 277-300, LNM 2078 (2013)

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 times

Nature: Original