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
XVII: 18, 179-184, LNM 986 (1983)
HE, Sheng-Wu;
YAN, Jia-An;
ZHENG, Wei-An
Sur la convergence des semimartingales continues dans ${\bf R}^n$ et des martingales dans une variété (
Stochastic calculus,
Stochastic differential geometry)
Say that a continuous semimartingale $X$ with canonical decomposition $X_0+M+A$ converges perfectly on an event $E$ if both $M_t$ and $\int_0^t|dA_s|$ have an a.s. limit on $E$ when $t\rightarrow \infty $. It is established that if $A_t$ has the form $\int_0^tH_sd[M,M]_s$, $X$ converges perfectly on the event $\{\sup_t|X_t|+\lim\sup_tH_t <\infty \}$. A similar (but less simple) statement is shown for multidimensional $X$; and an application is given to martingales in manifolds: every point of a manifold $V$ (with a connection) has a neighbourhood $U$ such that, given any $V$-valued martingale $X$, almost all paths of $X$ that eventually remain in $U$ are convergent
Comment: The latter statement (martingale convergence) is very useful; more recent proofs use convex functions instead of perfect convergence. The next talk
1719 is a small remark on perfect convergence
Keywords: Semimartingales,
Martingales in manifoldsNature: Original Retrieve article from Numdam
XVII: 33, 346-348, LNM 986 (1983)
HE, Sheng-Wu
Some remarks on single jump processes Retrieve article from Numdam
XVII: 34, 349-352, LNM 986 (1983)
HE, Sheng-Wu
The representation of Poisson functionals Retrieve article from Numdam
XVIII: 15, 174-178, LNM 1059 (1984)
HE, Sheng-Wu;
ZHENG, Wei-An
Remarques sur la convergence des martingales dans les variétés Retrieve article from Numdam
XVIII: 22, 256-267, LNM 1059 (1984)
HE, Sheng-Wu;
WANG, Jia-Gang
Two results on jump processes Retrieve article from Numdam
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 Retrieve article from Numdam
XXIX: 19, 202-217, LNM 1613 (1995)
QIAN, Zhongmin;
HE, Sheng-Wu
On the hypercontractivity of Ornstein-Uhlenbeck semigroups with drift Retrieve article from Numdam
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
