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V: 16, 170-176, LNM 191 (1971)

**MEYER, Paul-André**

Sur un article de Dubins (Martingale theory)

Description of a Skorohod imbedding procedure for real valued r.v.'s due to Dubins (*Ann. Math. Stat.*, **39,** 1968), using a remarkable discrete approximation of measures. It does not use randomization

Comment: This beautiful method to realize Skorohod's imbedding is related to that of Chacon and Walsh in 1002. For a deeper study see Bretagnolle 802. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Exposition

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V: 23, 237-250, LNM 191 (1971)

**MEYER, Paul-André**

Travaux de H. Rost en théorie du balayage (Potential theory, Ergodic theory)

The ``filling scheme'' is a technique used in ergodic theory to prove Hopf's maximal Lemma and the Chacon-Ornstein theorem, studied in detail by H.~Rost (*Zeit. für W-theorie,* **15**, 1970; *Ann. Inst. Fourier,* **21**, 1971): it provides a solution to Skorohod's imbedding problem for measures on discrete time Markov processes. Here it is also used to prove Brunel's Lemma in pointwise ergodic theory

Comment: Extension to continuous time in Meyer 612. See also 806, 1012. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Filling scheme, Brunel's lemma, Skorohod imbedding

Nature: Exposition, Original additions

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VI: 12, 130-150, LNM 258 (1972)

**MEYER, Paul-André**

Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)

The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (*Invent. Math.,* **14**, 1971) the construction is extended to continuous time Markov processes. In the transient case, the results are translated in potential-theoretic language, and proved using techniques due to Mokobodzki. Then the general case follows from this result applied to a space-time extension of the semi-group

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Filling scheme, Balayage of measures, Skorohod imbedding

Nature: Exposition, Original additions

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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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VIII: 02, 11-19, LNM 381 (1974)

**BRETAGNOLLE, Jean**

Une remarque sur le problème de Skorohod (Brownian motion)

The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see 516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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VIII: 06, 27-36, LNM 381 (1974)

**DINGES, Hermann**

Stopping sequences (Markov processes, Potential theory)

Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process

Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see 523, 524, 612. This article contains announcements of further results

Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme

Nature: Original

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VIII: 11, 150-154, LNM 381 (1974)

**HEATH, David C.**

Skorohod stopping via Potential Theory (Potential theory, Markov processes)

The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $**R**^n$ is given. A theorem of Mokobodzki on réduites is used as an important technical tool

Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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IX: 31, 515-517, LNM 465 (1975)

**HEATH, David C.**

Skorohod stopping in discrete time (Markov processes, Potential theory)

Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times

Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath 811. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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X: 02, 19-23, LNM 511 (1976)

**CHACON, Rafael V.**; **WALSH, John B.**

One-dimensional potential imbedding (Brownian motion)

The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool

Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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X: 12, 194-208, LNM 511 (1976)

**ROST, Hermann**

Skorohod stopping times of minimal variance (Markov processes)

Root's (*Ann. Math. Stat.*, **40**, 1969) solution of the Shorohod imbedding problem for Brownian motion uses the hitting time of a barrier in space-time. Here Root's construction is extended to general Markov processes, an optimality property of Root's stopping times is proved, as well as the uniqueness of such stopping times

Comment: For previous work of the author on Skorohod's imbedding see*Ann. M. Stat.* **40**, 1969 and *Invent. Math.* **14**, 1971, and in this Seminar 523, 613, 806. A general survey on the Skorohod embedding problem is Ob\lój, *Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XIII: 06, 90-115, LNM 721 (1979)

**AZÉMA, Jacques**; **YOR, Marc**

Une solution simple au problème de Skorokhod (Brownian motion)

An explicit solution is given to Skorohod's problem: given a distribution $\mu$ with mean $0$ and finite second moment $\sigma^2$, find a (non randomized) stopping time $T$ of a Brownian motion $(X_t)$ such that $X_T$ has the distribution $\mu$ and $E[T]=\sigma^2$. It is shown that if $S_t$ is the one-sided supremum of $X$ at time $t$, $T=\inf\{t:S_t\ge\psi(X_t)\}$ solves the problem, where $\psi(x)$ is the barycenter of $\mu$ restricted to $[x,\infty[$. The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray-Knight theorem on local times

Comment: The subject is further investigated in 1356 and 1441. See also 1515. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XIII: 56, 625-633, LNM 721 (1979)

**AZÉMA, Jacques**; **YOR, Marc**

Le problème de Skorokhod~: compléments à l'exposé précédent (Brownian motion)

What the title calls ``the preceding talk'' is 1306. The method is extended to (centered) measures possessing a moment of order one instead of two, preserving the uniform integrability of the stopped martingale

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XIV: 40, 357-391, LNM 784 (1980)

**FALKNER, Neil**

On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times (Brownian motion, Potential theory)

The problem discussed here is the Skorohod representation of a measure $\nu$ as the distribution of $B_T$, where $(B_t)$ is Brownian motion in $**R**^n$ with the initial measure $\mu$, and $T$ is a *non-randomized * stopping time. The conditions given are sufficient in all cases, necessary if $\mu$ does not charge polar sets

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XIV: 41, 392-396, LNM 784 (1980)

**PIERRE, Michel**

Le problème de Skorohod~: une remarque sur la démonstration d'Azéma-Yor (Brownian motion)

This is an addition to 1306 and 1356, showing how the proof can be reduced to that of a regular case, where it becomes simpler

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XV: 16, 227-250, LNM 850 (1981)

**ROGERS, L.C.G.**

Williams' characterization of the Brownian excursion law: proof and applications (Brownian motion)

In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams*Diffusions, Markov Processes and Martingales,* Wiley 1979, II.67), and it collects a number of applications, including the Azéma-Yor approach to Skorohod's imbedding theorem (1306)

Keywords: Excursions, Explicit laws, Bessel processes, Skorohod imbedding

Nature: Original

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XVII: 24, 221-224, LNM 986 (1983)

**BASS, Richard F.**

Skorohod imbedding via stochastic integrals (Brownian motion)

A centered probability $\mu$ on $\bf R$ is the law of $g(X_1)$, for a suitable function $g$ and $(X_t,\ t\le 1)$ a Brownian motion. The martingale with terminal value $g(X_1)$ is a time change $(T(t), \ t\le1)$ of a Brownian motion $\beta$; it is shown that $T(1)$ is a stopping time for $\beta$, thus showing the Skorohod embedding for $\mu$

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XVII: 25, 225-226, LNM 986 (1983)

**MEILIJSON, Isaac**

On the Azéma-Yor stopping time (Brownian motion)

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XVII: 26, 227-239, LNM 986 (1983)

**VALLOIS, Pierre**

Le problème de Skorokhod sur $\bf R$ : une approche avec le temps local (Brownian motion)

A solution to Skorohod's embedding problem is given, using the first hiting time of a set by the 2-dimensional process which consists of Brownian motion and its local time at zero. The author's aim is to ``correct'' the asymmetry inherent to the Azéma-Yor construction

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding, Local times

Nature: Original

Retrieve article from Numdam

Sur un article de Dubins (Martingale theory)

Description of a Skorohod imbedding procedure for real valued r.v.'s due to Dubins (

Comment: This beautiful method to realize Skorohod's imbedding is related to that of Chacon and Walsh in 1002. For a deeper study see Bretagnolle 802. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Exposition

Retrieve article from Numdam

V: 23, 237-250, LNM 191 (1971)

Travaux de H. Rost en théorie du balayage (Potential theory, Ergodic theory)

The ``filling scheme'' is a technique used in ergodic theory to prove Hopf's maximal Lemma and the Chacon-Ornstein theorem, studied in detail by H.~Rost (

Comment: Extension to continuous time in Meyer 612. See also 806, 1012. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Filling scheme, Brunel's lemma, Skorohod imbedding

Nature: Exposition, Original additions

Retrieve article from Numdam

VI: 12, 130-150, LNM 258 (1972)

Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)

The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Filling scheme, Balayage of measures, Skorohod imbedding

Nature: Exposition, Original additions

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

VIII: 02, 11-19, LNM 381 (1974)

Une remarque sur le problème de Skorohod (Brownian motion)

The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see 516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

VIII: 06, 27-36, LNM 381 (1974)

Stopping sequences (Markov processes, Potential theory)

Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process

Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see 523, 524, 612. This article contains announcements of further results

Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme

Nature: Original

Retrieve article from Numdam

VIII: 11, 150-154, LNM 381 (1974)

Skorohod stopping via Potential Theory (Potential theory, Markov processes)

The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $

Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

IX: 31, 515-517, LNM 465 (1975)

Skorohod stopping in discrete time (Markov processes, Potential theory)

Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times

Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath 811. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

X: 02, 19-23, LNM 511 (1976)

One-dimensional potential imbedding (Brownian motion)

The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool

Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

X: 12, 194-208, LNM 511 (1976)

Skorohod stopping times of minimal variance (Markov processes)

Root's (

Comment: For previous work of the author on Skorohod's imbedding see

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIII: 06, 90-115, LNM 721 (1979)

Une solution simple au problème de Skorokhod (Brownian motion)

An explicit solution is given to Skorohod's problem: given a distribution $\mu$ with mean $0$ and finite second moment $\sigma^2$, find a (non randomized) stopping time $T$ of a Brownian motion $(X_t)$ such that $X_T$ has the distribution $\mu$ and $E[T]=\sigma^2$. It is shown that if $S_t$ is the one-sided supremum of $X$ at time $t$, $T=\inf\{t:S_t\ge\psi(X_t)\}$ solves the problem, where $\psi(x)$ is the barycenter of $\mu$ restricted to $[x,\infty[$. The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray-Knight theorem on local times

Comment: The subject is further investigated in 1356 and 1441. See also 1515. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIII: 56, 625-633, LNM 721 (1979)

Le problème de Skorokhod~: compléments à l'exposé précédent (Brownian motion)

What the title calls ``the preceding talk'' is 1306. The method is extended to (centered) measures possessing a moment of order one instead of two, preserving the uniform integrability of the stopped martingale

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIV: 40, 357-391, LNM 784 (1980)

On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times (Brownian motion, Potential theory)

The problem discussed here is the Skorohod representation of a measure $\nu$ as the distribution of $B_T$, where $(B_t)$ is Brownian motion in $

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIV: 41, 392-396, LNM 784 (1980)

Le problème de Skorohod~: une remarque sur la démonstration d'Azéma-Yor (Brownian motion)

This is an addition to 1306 and 1356, showing how the proof can be reduced to that of a regular case, where it becomes simpler

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XV: 16, 227-250, LNM 850 (1981)

Williams' characterization of the Brownian excursion law: proof and applications (Brownian motion)

In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams

Keywords: Excursions, Explicit laws, Bessel processes, Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XVII: 24, 221-224, LNM 986 (1983)

Skorohod imbedding via stochastic integrals (Brownian motion)

A centered probability $\mu$ on $\bf R$ is the law of $g(X_1)$, for a suitable function $g$ and $(X_t,\ t\le 1)$ a Brownian motion. The martingale with terminal value $g(X_1)$ is a time change $(T(t), \ t\le1)$ of a Brownian motion $\beta$; it is shown that $T(1)$ is a stopping time for $\beta$, thus showing the Skorohod embedding for $\mu$

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XVII: 25, 225-226, LNM 986 (1983)

On the Azéma-Yor stopping time (Brownian motion)

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XVII: 26, 227-239, LNM 986 (1983)

Le problème de Skorokhod sur $\bf R$ : une approche avec le temps local (Brownian motion)

A solution to Skorohod's embedding problem is given, using the first hiting time of a set by the 2-dimensional process which consists of Brownian motion and its local time at zero. The author's aim is to ``correct'' the asymmetry inherent to the Azéma-Yor construction

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding, Local times

Nature: Original

Retrieve article from Numdam