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
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