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VI: 11, 118-129, LNM 258 (1972)

**MEYER, Paul-André**

La mesure de H. Föllmer en théorie des surmartingales (Martingale theory)

The Föllmer measure of a supermartingale is an extension to very general situation of the construction of $h$-path processes in the Markovian case. Let $\Omega$ be a probability space with a filtration, let $\Omega'$ be the product space $[0,\infty]\times\Omega$, the added coordinate playing the role of a lifetime $\zeta$. Then the Föllmer measure associated with a supermartingale $(X_t)$ is a measure $\mu$ on this enlarged space which satisfies the property $\mu(]T,\infty])=E(X_T)$ for any stopping time $T$, and simple additional properties to ensure uniqueness. When $X_t$ is a class (D) potential, it turns out to be the usual Doléans measure, but except in this case its existence requires some measure theoretic conditions on $\Omega$; which are slightly different here from those used by Föllmer,*Zeit für X-theorie,* **21**, 1970

Keywords: Supermartingales, Föllmer measures

Nature: Exposition, Original additions

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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.*, **2**

Keywords: Quasimartingales, Föllmer measures

Nature: Original

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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 1622

Keywords: Stochastic exponentials, Multiplicative decomposition, Angle bracket, Girsanov's theorem, Föllmer measures

Nature: Original

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La mesure de H. Föllmer en théorie des surmartingales (Martingale theory)

The Föllmer measure of a supermartingale is an extension to very general situation of the construction of $h$-path processes in the Markovian case. Let $\Omega$ be a probability space with a filtration, let $\Omega'$ be the product space $[0,\infty]\times\Omega$, the added coordinate playing the role of a lifetime $\zeta$. Then the Föllmer measure associated with a supermartingale $(X_t)$ is a measure $\mu$ on this enlarged space which satisfies the property $\mu(]T,\infty])=E(X_T)$ for any stopping time $T$, and simple additional properties to ensure uniqueness. When $X_t$ is a class (D) potential, it turns out to be the usual Doléans measure, but except in this case its existence requires some measure theoretic conditions on $\Omega$; which are slightly different here from those used by Föllmer,

Keywords: Supermartingales, Föllmer measures

Nature: Exposition, Original additions

Retrieve article from Numdam

IX: 19, 408-419, LNM 465 (1975)

Mesure de Föllmer en théorie des quasimartingales (Martingale theory)

The Föllmer measure associated with a positive supermartingale, or more generally a quasimartingale (Föllmer,

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,

Keywords: Quasimartingales, Föllmer measures

Nature: Original

Retrieve article from Numdam

X: 21, 432-480, LNM 511 (1976)

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

Keywords: Stochastic exponentials, Multiplicative decomposition, Angle bracket, Girsanov's theorem, Föllmer measures

Nature: Original

Retrieve article from Numdam