X: 20, 422-431, LNM 511 (1976)
YAN, Jia-An;
YOEURP, Chantha
Représentation des martingales comme intégrales stochastiques des processus optionnels (
Martingale theory,
Stochastic calculus)
An attempt to build a theory similar to the previsible representation property with respect to a basic local martingale, but using the optional stochastic integral instead of the standard one
Comment: Apparently this ``optional representation property'' has not been used since
Keywords: Optional stochastic integralsNature: Original
Retrieve article from Numdam
X: 21, 432-480, LNM 511 (1976)
YOEURP, Chantha
Décomposition des martingales locales et formules exponentielles (
Martingale theory,
Stochastic calculus)
It is shown that local martingales can be decomposed uniquely into three pieces, a continuous part and two purely discontinuous pieces, one with accessible jumps, and one with totally inaccessible jumps. Two beautiful lemmas say that a purely discontinuous local martingale whose jumps are summable is a finite variation process, and if it has accessible jumps, then it is the sum of its jumps without compensation. Conditions are given for the existence of the angle bracket of two local martingales which are not locally square integrable. Lemma 2.3 is the lemma often quoted as ``Yoeurp's Lemma'': given a local martingale $M$ and a previsible process of finite variation $A$, $[M,A]$ is a local martingale. The definition of a local martingale on an open interval $[0,T[$ is given when $T$ is previsible, and the behaviour of local martingales under changes of laws (Girsanov's theorem) is studied in a set up where the positive martingale defining the mutual density is replaced by a local martingale. The existence and uniqueness of solutions of the equation $Z_t=1+\int_0^t\tilde Z_s dX_s$, where $X$ is a given special semimartingale of decomposition $M+A$, and $\widetilde Z$ is the previsible projection of the unknown special semimartingale $Z$, is proved under an assumption that the jumps $ėlta A_t$ do not assume the value $1$. Then this ``exponential'' is used to study the multiplicative decomposition of a positive supermartingale in full generality
Comment: The problems in this paper have some relation with Kunita
1005 (in a Markovian set up), and are further studied by Yoeurp in LN
1118,
Grossissements de filtrations, 1985. The subject of multiplicative decompositions of positive submartingales is much more difficult since they may vanish. For a simple case see in this volume Yoeurp-Meyer
1023. The general case is due to Azéma (
Z. für W-theorie, 45, 1978, presented in
1321) See also
1622Keywords: Stochastic exponentials,
Multiplicative decomposition,
Angle bracket,
Girsanov's theorem,
Föllmer measuresNature: Original Retrieve article from Numdam
X: 23, 501-504, LNM 511 (1976)
MEYER, Paul-André;
YOEURP, Chantha
Sur la décomposition multiplicative des sousmartingales positives (
Martingale theory)
This paper expands part of Yoeurp's paper
1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$
Comment: See the comments on
1021 for the general case. The latter result is related to Meyer
817. For a related paper, see
1203. Further study in
1620Keywords: Multiplicative decompositionNature: Original Retrieve article from Numdam
XIII: 08, 118-125, LNM 721 (1979)
YOEURP, Chantha
Sauts additifs et sauts multiplicatifs des semi-martingales (
Martingale theory,
General theory of processes)
First of all, the jump processes of special semimartingales are characterized (using a result of
1121,
1129 on the jump processes of local martingales). Then a similar problem is solved for multiplicative jumps, a result which includes that of Garcia and al.
1206. A technical lemma characterizes optional processes, bounded in $L^1$, whose previsible projection vanishes
Keywords: Jump processesNature: Original Retrieve article from Numdam
XIII: 53, 614-619, LNM 721 (1979)
YOEURP, Chantha
Solution explicite de l'équation $Z_t=1+\int_0^t |Z_{s-}|\,dX_s$ (
Stochastic calculus)
The title describes completely the paper
Keywords: Stochastic differential equationsNature: Original Retrieve article from Numdam
XIV: 28, 249-253, LNM 784 (1980)
YOEURP, Chantha
Sur la dérivation des intégrales stochastiques (
Stochastic calculus)
The following problem is discussed: under which conditions do ratios of the form $\int_t^{t+h} H_s\,dM_s/(M_{t+h}-M_t)$ converge to $H_t$ as $h\rightarrow 0$? It is shown that positive results due to Isaacson (
Ann. Math. Stat. 40, 1979) in the Brownian case fail in more general situations
Comment: See also
1529Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XIV: 29, 254-254, LNM 784 (1980)
YOEURP, Chantha
Rectificatif à l'exposé de C.S. Chou (
Stochastic calculus)
A mistake in the proof of
1337 is corrected, the result remaining true without additional assumptions
Keywords: Local times,
Semimartingales,
JumpsNature: Correction Retrieve article from Numdam
XV: 29, 399-412, LNM 850 (1981)
YOEURP, Chantha
Sur la dérivation stochastique au sens de Davis (
Stochastic calculus,
Brownian motion)
The problem is that of estimating $H_t$ from a knowledge of the process $\int H_sdX_s$, where $X_t$ is a continuous (semi)martingale, as a limit of ratios of the form $\int_t^{t+\alpha} H_sdX_s/(X_{t+\alpha}-X_t)$, or replacing $t+\alpha$ by suitable stopping times
Comment: The problem was suggested and partially solved by Mark H.A. Davis (
Teor. Ver. Prim.,
20, 1975, 887--892). See also
1428Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XVI: 20, 234-237, LNM 920 (1982)
YOEURP, Chantha
Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (
Brownian motion,
Stochastic calculus)
A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$
Comment: See
1023,
1321Keywords: Multiplicative decomposition,
Change of variable formula,
Local timesNature: Original Retrieve article from Numdam
