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XV: 43, 627-631, LNM 850 (1981)

**WANG, Jia-Gang**

Some remarks on processes with independent increments (Independent increments)

This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also 1544

Keywords: Poisson processes, Lévy measures

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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XVIII: 22, 256-267, LNM 1059 (1984)

**HE, Sheng-Wu**; **WANG, Jia-Gang**

Two results on jump processes

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XXII: 26, 260-270, LNM 1321 (1988)

**HE, Sheng-Wu**; **WANG, Jia-Gang**

Remarks on absolute continuity, contiguity and convergence in variation of probability measures

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XXXI: 09, 80-84, LNM 1655 (1997)

**HE, Sheng-Wu**; **WANG, Jia-Gang**

The hypercontractivity of Ornstein-Uhlenbeck semigroups with drift, revisited

Retrieve article from Numdam

Some remarks on processes with independent increments (Independent increments)

This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also 1544

Keywords: Poisson processes, Lévy measures

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

XVIII: 22, 256-267, LNM 1059 (1984)

Two results on jump processes

Retrieve article from Numdam

XXII: 26, 260-270, LNM 1321 (1988)

Remarks on absolute continuity, contiguity and convergence in variation of probability measures

Retrieve article from Numdam

XXXI: 09, 80-84, LNM 1655 (1997)

The hypercontractivity of Ornstein-Uhlenbeck semigroups with drift, revisited

Retrieve article from Numdam