IX: 19, 408-419, LNM 465 (1975)
STRICKER, Christophe
Mesure de Föllmer en théorie des quasimartingales (
Martingale theory)
The Föllmer measure associated with a positive supermartingale, or more generally a quasimartingale (Föllmer,
Z. für W-theorie, 21, 1972;
Ann. Prob. 1, 1973) is constructed using a weak limit procedure instead of a projective limit
Comment: On Föllmer measures see
611. This paper corresponds to an early stage in the theory of quasimartingales, for which the main reference was Orey,
Proc. Fifth Berkeley Symp.,
2Keywords: Quasimartingales,
Föllmer measuresNature: Original
Retrieve article from Numdam
IX: 20, 420-424, LNM 465 (1975)
STRICKER, Christophe
Une caractérisation des quasimartingales (
Martingale theory)
An integral criterion is shown to be equivalent to the usual definition of a quasimartingale using the stochastic variation
Keywords: QuasimartingalesNature: Original Retrieve article from Numdam
XI: 23, 365-375, LNM 581 (1977)
DELLACHERIE, Claude;
STRICKER, Christophe
Changements de temps et intégrales stochastiques (
Martingale theory)
A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see
925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also
1135Comment: The last section states an interesting open problem
Keywords: Changes of time,
Spectral representationNature: Original Retrieve article from Numdam
XII: 02, 20-21, LNM 649 (1978)
STRICKER, Christophe
Une remarque sur les changements de temps et les martingales locales (
Martingale theory)
It is well known (see
606) that in general the class of local martingales is not invariant under changes of time. Here it is shown that, if ${\cal F}_0$ is trivial, a process which remains a local martingale under all changes of time (with bounded stopping times) is a true martingale (in full generality, it is so conditionally to ${\cal F}_0$)
Keywords: Changes of time,
Weak martingalesNature: Original Retrieve article from Numdam
XII: 25, 364-377, LNM 649 (1978)
STRICKER, Christophe
Les ralentissements en théorie générale des processus (
General theory of processes)
Given a filtration $({\cal F}t)$ and a stopping time $T$, we may define a new filtration $({\cal G}_t)$ as follows: we introduce an independent random variable $S$, and in intuitive language, we run the picture of $({\cal F}_t)$ up to time $T$, freeze the image between times $T$ and $T+S$, and then start running it again. The main result of this paper is the possibility, by performing this at all the times of discontinuity of $({\cal F}_t)$, to construct a filtration $({\cal G}_t)$ which is quasi-left-continuous. Though the idea is simple, there are considerable technical difficulties
Nature: Original Retrieve article from Numdam
XIII: 07, 116-117, LNM 721 (1979)
ÉMERY, Michel;
STRICKER, Christophe
Démonstration élémentaire d'un résultat d'Azéma et Jeulin (
Martingale theory)
A short proof is given for the following result: given a positive supermartingale $X$ and $h>0$, the supermartingale $E[X_{t+h}\,|\,{\cal F}_t]$ belongs to the class (D). The original proof (
Ann. Inst. Henri Poincaré, 12, 1976) used Föllmer's measures
Keywords: Class (D) processesNature: Original Retrieve article from Numdam
XIII: 19, 233-237, LNM 721 (1979)
STRICKER, Christophe
Sur la $p$-variation des surmartingales (
Martingale theory)
The method of the preceding paper of Bruneau
1318 is extended to all right-continuous semimartingales
Keywords: $p$-variation,
UpcrossingsNature: Original Retrieve article from Numdam
XIII: 20, 238-239, LNM 721 (1979)
STRICKER, Christophe
Une remarque sur l'exposé précédent (
Martingale theory)
A few comments are added to the preceding paper
1319, concerning in particular its relationship with results of Lépingle,
Zeit. für W-Theorie, 36, 1976
Keywords: $p$-variation,
UpcrossingsNature: Original Retrieve article from Numdam
XIII: 40, 472-477, LNM 721 (1979)
STRICKER, Christophe
Semimartingales et valeur absolue (
General theory of processes)
For the general notation, see
1338. A result of Yoeurp that absolute values preserves quasimartingales is extended: convex functions satisfying a Lipschitz condition operate on quasimartingales. For $p\ge1$, $X\in H^p$ implies $|X|^p\in H^1$. Then it is shown that for a continuous adapted process $X$, it is equivalent to say that $X$ and $|X|$ are quasimartingales (or semimartingales). Then comes a result related to the main problem of this series: with the general notations above, if $X$ is assumed to be a quasimartingale such that $X_{D_t}=0$ for all $t$, if the process $Z$ is progressive and bounded, then the process $Z_{g_t}X_t$ is a quasimartingale
Comment: A complement is given in the next paper
1341. See also
1351Keywords: Balayage,
QuasimartingalesNature: Original Retrieve article from Numdam
XIII: 41, 478-487, LNM 721 (1979)
MEYER, Paul-André;
STRICKER, Christophe;
YOR, Marc
Sur une formule de la théorie du balayage (
General theory of processes)
For the notation, see the review of
1340. It is shown here that under the same hypotheses, the semimartingale $Z_{g_t}X_t$ is a sum of three terms: the stochastic integral $\int_0^t \zeta_s dX_s$, where $\zeta$ is the previsible projection of $Z$, an explicit sum of jumps involving $Z-\zeta$, and a mysterious continuous process with finite variation $(R_t)$ such that $dR_t$ is carried by $H$, equal to $0$ if $Z$ was optional
Comment: See
1351,
1357Keywords: Balayage,
Balayage formulaNature: Original Retrieve article from Numdam
XIII: 51, 610-610, LNM 721 (1979)
STRICKER, Christophe
Encore une remarque sur la ``formule de balayage'' (
General theory of processes)
A slight extension of
1341Keywords: BalayageNature: Original Retrieve article from Numdam
XIV: 10, 104-111, LNM 784 (1980)
STRICKER, Christophe
Prolongement des semi-martingales (
Stochastic calculus)
The problem consists in characterizing semimartingales on $]0,\infty[$ which can be ``closed at infinity'', and the similar problem at $0$. The criteria are similar to the Vitali-Hahn-Saks theorem and involve convergence in probability of suitable stochastic integrals. The proof rests on a functional analytic result of Maurey-Pisier
Keywords: Semimartingales,
Semimartingales in an open intervalNature: Original Retrieve article from Numdam
XIV: 11, 112-115, LNM 784 (1980)
STRICKER, Christophe
Projection optionnelle des semi-martingales (
Stochastic calculus)
Let $({\cal G}_t)$ be a subfiltration of $({\cal F}_t)$. Since the optional projection on $({\cal G}_t)$ of a ${\cal F}$-martingale is a ${\cal G}$-martingale, and the projection of an increasing process a ${\cal G}$-submartingale, projections of ${\cal F}$-semimartingales ``should be'' ${\cal G}$-semimartingales. This is true for quasimartingales, but false in general
Comment: The main results on subfiltrations are proved by Stricker in
Zeit. für W-Theorie, 39, 1977
Keywords: Semimartingales,
Projection theoremsNature: 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: 31, 490-492, LNM 850 (1981)
STRICKER, Christophe
Sur deux questions posées par Schwartz (
Stochastic calculus)
Schwartz studied semimartingales in random open sets, and raised two questions: Given a semimartingale $X$ and a random open set $A$, 1) Assume $X$ is increasing in every subinterval of $A$; then is $X$ equal on $A$ to an increasing adapted process on the whole line? 2) Same statement with ``increasing'' replaced by ``continuous''. Schwartz could prove statement 1) assuming $X$ was continuous. It is proved here that 1) is false if $X$ is only cadlag, and that 2) is false in general, though it is true if $A$ is previsible, or only accessible
Keywords: Random sets,
Semimartingales in a random open setNature: Original Retrieve article from Numdam
XV: 32, 493-498, LNM 850 (1981)
STRICKER, Christophe
Quasi-martingales et variations (
Martingale theory)
This paper contains remarks on quasimartingales, the most useful of which being perhaps the fact that, for a right-continuous process, the stochastic variation is the same with respect to the filtrations $({\cal F}_{t})$ and $({\cal F}_{t-})$
Keywords: QuasimartingalesNature: Original Retrieve article from Numdam
XV: 33, 499-522, LNM 850 (1981)
STRICKER, Christophe
Quelques remarques sur la topologie des semimartingales. Applications aux intégrales stochastiques (
Stochastic calculus)
This paper contains a number of useful technical results on the topology of semimartingales (see
1324), some of which were previously known with more complicated proofs. In particular, it is shown how to improve the convergence of sequences of semimartingales by a convenient change of probability. The topology of semimartingales is used to handle elegantly the stochastic integration of previsible processes which are not locally bounded (see
1415). Finally, boundedness of a set of semimartingales is shown to be equivalent to the boundedness (in an elementary sense) of a set of increasing processes controlling them in the sense of Métivier-Pellaumail (see
1412,
1413,
1414)
Keywords: Semimartingales,
Stochastic integrals,
Spaces of semimartingales,
Métivier-Pellaumail inequalityNature: Original Retrieve article from Numdam
XV: 34, 523-525, LNM 850 (1981)
STRICKER, Christophe
Sur la caractérisation des semi-martingales (
General theory of processes,
Stochastic calculus)
This is a sequel to the preceding paper
1533, giving a simple proof that any semimartingale may be brought into any class ${\cal S}^p$ by a convenient change of probability
Keywords: Semimartingales,
Spaces of semimartingalesNature: Original Retrieve article from Numdam
XVI: 17, 213-218, LNM 920 (1982)
FALKNER, Neil;
STRICKER, Christophe;
YOR, Marc
Temps d'arrêt riches et applications (
General theory of processes)
This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see
1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper
1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$
Keywords: Stopping times,
Local times,
Semimartingales,
Previsible processesNature: Original Retrieve article from Numdam
XVI: 18, 219-220, LNM 920 (1982)
STRICKER, Christophe
Les intervalles de constance de $\langle X,X\rangle$ (
Martingale theory,
Stochastic calculus)
For a continuous (local) martingale $X$, the constancy intervals of $X$ and $<X,X>$ are exactly the same. What about general local martingales? It is proved that $X$ is constant on the constancy intervals of $<X,X>$, and the converse holds if $X$ has the previsible representation property
Keywords: Quadratic variation,
Previsible representationNature: Original Retrieve article from Numdam
XVII: 29, 298-305, LNM 986 (1983)
ABOULAÏCH, Rajae;
STRICKER, Christophe
Variation des processus mesurables Retrieve article from Numdam
XVII: 30, 306-310, LNM 986 (1983)
ABOULAÏCH, Rajae;
STRICKER, Christophe
Sur un théorème de Talagrand Retrieve article from Numdam
XVIII: 11, 144-147, LNM 1059 (1984)
STRICKER, Christophe
Approximation du crochet de certaines semimartingales continues Retrieve article from Numdam
XVIII: 12, 148-153, LNM 1059 (1984)
STRICKER, Christophe
Caractérisation des semimartingales Retrieve article from Numdam
XIX: 14, 209-217, LNM 1123 (1985)
STRICKER, Christophe
Lois de semimartingales et critères de compacité Retrieve article from Numdam
XIX: 15, 218-221, LNM 1123 (1985)
STRICKER, Christophe
Une remarque sur une certaine classe de semimartingales Retrieve article from Numdam
XX: 04, 34-39, LNM 1204 (1986)
PONTIER, Monique;
STRICKER, Christophe;
SZPIRGLAS, Jacques
Sur le théorème de représentation par rapport à l'innovation Retrieve article from Numdam
XXII: 13, 144-146, LNM 1321 (1988)
STRICKER, Christophe
À propos d'une conjecture de Meyer Retrieve article from Numdam
XXIV: 16, 266-274, LNM 1426 (1990)
ANSEL, Jean-Pascal;
STRICKER, Christophe
Quelques remarques sur un théorème de Yan Retrieve article from Numdam
XXVII: 03, 22-29, LNM 1557 (1993)
ANSEL, Jean-Pascal;
STRICKER, Christophe
Unicité et existence de la loi minimale Retrieve article from Numdam
XXVII: 04, 30-32, LNM 1557 (1993)
ANSEL, Jean-Pascal;
STRICKER, Christophe
Décomposition de Kunita-Watanabe Retrieve article from Numdam
XXVIII: 18, 189-194, LNM 1583 (1994)
MONAT, Pascale;
STRICKER, Christophe
Fermeture de $G_T(\Theta)$ et de $L^2({\cal F}_0)+G_T(\Theta)$ Retrieve article from Numdam
XXX: 02, 12-23, LNM 1626 (1996)
CHOULLI, Tahir;
STRICKER, Christophe
Deux applications de la décomposition de Galtchouk-Kunita-Watanabe Retrieve article from Numdam
XXXII: 06, 56-66, LNM 1686 (1998)
STRICKER, Christophe;
YAN, Jia-An
Some remarks on the optional decomposition theorem Retrieve article from Numdam
XXXII: 07, 67-72, LNM 1686 (1998)
CHOULLI, Tahir;
STRICKER, Christophe
Séparation d'une sur- et d'une sous-martingale par une martingale Retrieve article from Numdam
XXXV: 08, 139-148, LNM 1755 (2001)
KABANOV, Youri;
STRICKER, Christophe
On equivalent martingale measures with bounded densities Retrieve article from Numdam
XXXV: 09, 149-152, LNM 1755 (2001)
KABANOV, Youri;
STRICKER, Christophe
A teacher's note on no-arbitrage criteria Retrieve article from Numdam
XXXVI: 20, 413-414, LNM 1801 (2003)
STRICKER, Christophe
On the true submartingale property, d'après Schachermayer
XXXVI: 21, 415-418, LNM 1801 (2003)
STRICKER, Christophe
Simple strategies in exponential utility maximization
XXXVIII: 13, 186-194, LNM 1857 (2005)
KABANOV, Yuri;
STRICKER, Christophe
Remarks on the true no-arbitrage property
XXXIX: 11, 209-213, LNM 1874 (2006)
KABANOV, Yuri;
STRICKER, Christophe
The Dalang--Morton--Willinger theorem under delayed and restricted information
XLI: 21, 439-442, LNM 1934 (2008)
KABANOV, Yuri;
STRICKER, Christophe
On martingale selectors of cone-valued processes (
Theory of martingales)
Nature: Original
