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VI: 22, 233-242, LNM 258 (1972)

**WALSH, John B.**

The perfection of multiplicative functionals (Markov processes)

In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see 203) that every functional has a perfect modification, see also Dellacherie 304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure

Keywords: Multiplicative functionals, Perfection, Essential topology

Nature: Original

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VII: 25, 273-283, LNM 321 (1973)

**PINSKY, Mark A.**

Fonctionnelles multiplicatives opératrices (Markov processes)

This paper presents results due to the author (*Advances in Probability,* **3**, 1973). The essential idea is to consider multiplicative functionals of a Markov process taking values in the algebra of bounded operators of a Banach space $L$. Such a functional defines a semi-group acting on bounded $L$-valued functions. This semi-group determines the functional. The structure of functionals is investigated in the case of a finite Markov chain. The case where $L$ is finite dimensional and the Markov process is Browian motion is investigated too. Asymptotic results near $0$ are described

Comment: This paper explores the same idea as Jacod (*Mém. Soc. Math. France,* **35**, 1973), though in a very different way. See 816

Keywords: Multiplicative functionals, Multiplicative kernels

Nature: Exposition

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VIII: 01, 1-10, LNM 381 (1974)

**AZÉMA, Jacques**; **MEYER, Paul-André**

Une nouvelle représentation du type de Skorohod (Markov processes)

A Skorohod imbedding theorem for general Markov processes is proved, in which the stopping time is a randomized ``left'' terminal time. A uniqueness result is proved

Comment: The result is deduced from a representation of measures by left additive functionals, due to Azéma (*Invent. Math.* **18**, 1973 and this volume, 814). A general survey on the Skorohod embedding problem is Ob\lój, *Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding, Multiplicative functionals

Nature: Original

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The perfection of multiplicative functionals (Markov processes)

In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see 203) that every functional has a perfect modification, see also Dellacherie 304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure

Keywords: Multiplicative functionals, Perfection, Essential topology

Nature: Original

Retrieve article from Numdam

VII: 25, 273-283, LNM 321 (1973)

Fonctionnelles multiplicatives opératrices (Markov processes)

This paper presents results due to the author (

Comment: This paper explores the same idea as Jacod (

Keywords: Multiplicative functionals, Multiplicative kernels

Nature: Exposition

Retrieve article from Numdam

VIII: 01, 1-10, LNM 381 (1974)

Une nouvelle représentation du type de Skorohod (Markov processes)

A Skorohod imbedding theorem for general Markov processes is proved, in which the stopping time is a randomized ``left'' terminal time. A uniqueness result is proved

Comment: The result is deduced from a representation of measures by left additive functionals, due to Azéma (

Keywords: Skorohod imbedding, Multiplicative functionals

Nature: Original

Retrieve article from Numdam