IX: 04, 206-212, LNM 465 (1975)
CHOU, Ching Sung
Les inégalités des surmartingales d'après A.M. Garsia (
Martingale theory)
A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities
Keywords: Inequalities,
Burkholder inequalitiesNature: Exposition
Retrieve article from Numdam
IX: 05, 213-225, LNM 465 (1975)
CHOU, Ching Sung
Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (
Martingale theory)
The methods developed in discrete time by Garsia
Martingale Inequalities: Seminar Notes on Recent Progress, Benjamin, 1973, are extended to continuous time
Comment: See Lenglart-Lépingle-Pratelli
1404. These methods have now become standard, and can be found in a number of books
Keywords: Inequalities,
Burkholder inequalitiesNature: Original Retrieve article from Numdam
IX: 06, 226-236, LNM 465 (1975)
CHOU, Ching Sung;
MEYER, Paul-André
Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels (
General theory of processes)
Dellacherie has studied in
405 the filtration generated by a point process with one single jump. His study is extended here to the filtration generated by a discrete point process. It is shown in particular how to construct a martingale which has the previsible representation property
Comment: In spite or because of its simplicity, this paper has become a standard reference in the field. For a general account of the subject, see He-Wang-Yan,
Semimartingale Theory and Stochastic Calculus, CRC~Press 1992
Keywords: Point processes,
Previsible representationNature: Original Retrieve article from Numdam
XI: 21, 356-361, LNM 581 (1977)
CHOU, Ching Sung
Le processus des sauts d'une martingale locale (
Martingale theory)
Simple necessary and sufficient conditions are given on an optional process $\sigma_t$ (different from $0$ only at countably many stopping times) so that it is the process of jumps $ėlta M_t$ of some local martingale $M$
Comment: The same result is proved independently by D. Lépingle in this volume, see
1129. For an application see
1308, and for another approach
1335Keywords: Local martingales,
JumpsNature: Original Retrieve article from Numdam
XII: 14, 132-133, LNM 649 (1978)
CHOU, Ching Sung
Extension au cas continu d'un théorème de Dubins (
Martingale theory)
After a suitable translation, Dubins' theorem can be stated as follows: if $X$ is a positive submartingale with $X_t\in L^2$ for all $t$, then $X^2-[X,X]$ is a submartingale
Keywords: SubmartingalesNature: Original Retrieve article from Numdam
XIII: 22, 250-252, LNM 721 (1979)
CHOU, Ching Sung
Caractérisation d'une classe de semimartingales (
Martingale theory,
Stochastic calculus)
The class of semimartingales $X$ such that the stochastic integral $J\,
.\,X$ is a martingale for some nowhere vanishing previsible process $J$ is a natural class of martingale-like processes. Local martingales are exactly the members of this class which are special semimartingales
Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997)
Keywords: Local martingales,
Stochastic integralsNature: Original Retrieve article from Numdam
XIII: 37, 441-442, LNM 721 (1979)
CHOU, Ching Sung
Démonstration simple d'un résultat sur le temps local (
Stochastic calculus)
It follows from Ito's formula that the positive parts of those jumps of a semimartingale $X$ that originate below $0$ are summable. A direct proof is given of this fact
Comment: Though the idea is essentially correct, an embarrassing mistake is corrected as
1429Keywords: Local times,
Semimartingales,
JumpsNature: Original Retrieve article from Numdam
XIV: 12, 116-117, LNM 784 (1980)
CHOU, Ching Sung
Une caractérisation des semimartingales spéciales (
Stochastic calculus)
This is a useful addition to the next paper
1413: a semimartingale can be ``controlled'' (in the sense of Métivier-Pellaumail) by a locally integrable increasing process if and only if it is special
Comment: See also
1352Keywords: Semimartingales,
Métivier-Pellaumail inequality,
Special semimartingalesNature: Original Retrieve article from Numdam
XIV: 15, 128-139, LNM 784 (1980)
CHOU, Ching Sung;
MEYER, Paul-André;
STRICKER, Christophe
Sur l'intégrale stochastique de processus prévisibles non bornés (
Stochastic calculus)
The standard theory of stochastic integration deals with locally bounded previsible processes. The natural definition of the stochastic integral $H.X$ of a previsible process $H$ w.r.t. a semimartingale $X$ consists in assuming the existence of some decomposition $X=M+A$ such that $H.M$ exists in the martingale sense, and $H.A$ in the Stieltjes sense, and then defining $H.X$ as their sum. This turns out to be a very awkward definition. It is shown here to be equivalent to the following one: truncating $H$ at $n$, the standard stochastic integrals $H_n.X$ converge in the topology of semimartingales. This is clearly invariant under changes of law. A counterexample shows that integrability may be lost if the filtration is enlarged
Comment: See also
1417. This is a synthesis of earlier work, much of which is due to Jacod,
Calcul Stochastique et Problèmes de Martingales, Lect. Notes in M. 714. The contents of this paper appeared in book form in Dellacherie-Meyer,
Probabilités et Potentiel B, Chap. VIII, \S3. An equivalent definition is given by L. Schwartz in
1530, using the idea of ``formal semimartingales''. For further steps in the same direction, see Stricker
1533Keywords: Stochastic integralsNature: Exposition,
Original additions Retrieve article from Numdam
XV: 20, 285-289, LNM 850 (1981)
CHOU, Ching Sung
Une inégalité de martingales avec poids (
Martingale theory)
Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in
1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see
1326)
Keywords: Weighted norm inequalities,
Burkholder inequalities,
Moderate convex functionsNature: Original Retrieve article from Numdam
XVI: 37, 409-411, LNM 920 (1982)
CHOU, Ching Sung
Une remarque sur l'approximation des solutions d'e.d.s. Retrieve article from Numdam
XVII: 11, 117-120, LNM 986 (1983)
CHOU, Ching Sung
Sur certaines inégalités de théorie des martingales Retrieve article from Numdam
XVIII: 19, 219-222, LNM 1059 (1984)
CHOU, Ching Sung
Sur certaines généralisations de l'inégalité de Fefferman Retrieve article from Numdam
XXXV: 04, 87-97, LNM 1755 (2001)
CHAO, Tsung Ming;
CHOU, Ching Sung
Some remarks on the martingales satisfying the structure equation ${[X,X]}_t= t+\int_0^t\beta X_{s^-}\,dX_s$ Retrieve article from Numdam
