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VI: 05, 90-97, LNM 258 (1972)

**CHUNG, Kai Lai**

On universal field equations (General theory of processes)

There is a pun in the title, since ``field'' here is a $\sigma$-field and not a quantum field. The author proves useful results on the $\sigma$-fields ${\cal F}_{T-}$ and ${\cal F}_{T+}$ associated with an arbitrary random variable $T$ in the paper of Chung-Doob,*Amer. J. Math.*, **87**, 1965. As a corollary, he can prove easily that for a Hunt process, accessible = previsible

Keywords: Filtrations

Nature: Original

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XI: 22, 362-364, LNM 581 (1977)

**DELLACHERIE, Claude**

Sur la régularisation des surmartingales (Martingale theory)

It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)

Comment: See 1524

Keywords: General filtrations, Strong supermartingales

Nature: Original

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XII: 08, 57-60, LNM 649 (1978)

**MEYER, Paul-André**

Sur un théorème de J. Jacod (General theory of processes)

Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals

Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration

Keywords: Semimartingales, Enlargement of filtrations, Laws of semimartingales

Nature: Original

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XII: 09, 61-69, LNM 649 (1978)

**YOR, Marc**

Grossissement d'une filtration et semi-martingales~: théorèmes généraux (General theory of processes)

Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called*progressively enlarged * filtration is the smallest one $({\cal G}_t)$ containing $({\cal F}_t)$, and for which $L$ is a stopping time. The enlargement problem consists in describing the semimartingales $X$ of ${\cal F}$ which remain semimartingales in ${\cal G}$, and in computing their semimartingale characteristics. In this paper, it is proved that $X_tI_{\{t< L\}}$ is a semimartingale in full generality, and that $X_tI_{\{t\ge L\}}$ is a semimartingale whenever $L$ is *honest * for $\cal F$, i.e., is the end of an $\cal F$-optional set

Comment: This result was independently discovered by Barlow,*Zeit. für W-theorie,* 44, 1978, which also has a huge intersection with 1211. Complements are given in 1210, and an explicit decomposition formula for semimartingales in 1211

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 10, 70-77, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

A propos du travail de Yor sur le grossissement des tribus (General theory of processes)

This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 11, 78-97, LNM 649 (1978)

**JEULIN, Thierry**; **YOR, Marc**

Grossissement d'une filtration et semi-martingales~: Formules explicites (General theory of processes)

This contains very substantial improvements on 1209, namely, the explicit computation of the characteristics of the semimartingales involved

Comment: For additional results on enlargements, see the two Lecture Notes volumes**833** (T. Jeulin) and **1118**. See also 1350

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XIII: 34, 400-406, LNM 721 (1979)

**YOR, Marc**

Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)

This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$

Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$

Nature: Original

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XIII: 50, 574-609, LNM 721 (1979)

**JEULIN, Thierry**

Grossissement d'une filtration et applications (General theory of processes, Markov processes)

This is a sequel to the papers 1209 and 1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths

Keywords: Enlargement of filtrations, Williams decomposition

Nature: Original

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XIV: 20, 173-188, LNM 784 (1980)

**MEYER, Paul-André**

Les résultats de Jeulin sur le grossissement des tribus (General theory of processes, Stochastic calculus)

This is an introduction to beautiful results of Jeulin on enlargements, for which see*Zeit. für W-Theorie,* **52**, 1980, and above all the Lecture Notes vol. 833, *Semimartingales et grossissement d'une filtration *

Comment: See also 1329, 1350

Keywords: Enlargement of filtrations, Semimartingales

Nature: Exposition

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XIV: 21, 189-199, LNM 784 (1980)

**YOR, Marc**

Application d'un lemme de Jeulin au grossissement de la filtration brownienne (General theory of processes, Brownian motion)

The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment

Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')

Keywords: Enlargement of filtrations

Nature: Original

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XIV: 23, 205-208, LNM 784 (1980)

**SEYNOU, Aboubakary**

Sur la compatibilité temporelle d'une tribu et d'une filtration discrète (General theory of processes)

Let us say that a $\sigma$-field ${\cal G}$ is embedded into a filtration $({\cal H}_n)$ if ${\cal G}= {\cal H}_T$ for some ${\cal H}$-stopping time $T$, that a filtration $({\cal F}_n)$ is embedded into $({\cal H}_n)$ if ${\cal F}_S$ can be embedded into $({\cal H}_n)$ for every ${\cal F}$-stopping time $S$, and finally that ${\cal G}$ and ${\cal F}$ or $({\cal F}_n)$ are compatible if they can be embedded into one single filtration $({\cal H}_n)$. Then the question investigated is whether the compatibility of ${\cal G}$ and the individual $\sigma$-fields ${\cal F}_n$ implies that of ${\cal G}$ and the whole filtration $({\cal F}_n)$

Comment: This problem arose from the spectral point of view on stochastic integration as in 1123

Keywords: Filtrations

Nature: Original

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XV: 15, 210-226, LNM 850 (1981)

**JEULIN, Thierry**; **YOR, Marc**

Sur les distributions de certaines fonctionnelles du mouvement brownien (Brownian motion)

This paper gives new proofs and extensions of results due to Knight, concerning occupation times by the process $(S_t,B_t)$ up to time $T_a$, where $(B_t)$ is Brownian motion, $T_a$ the hitting time of $a$, and $(S_t)$ is $\sup_{s\le t} B_s$. The method uses enlargement of filtrations, and martingales similar to those of 1306. Theorem 3.7 is a decomposition of Brownian paths akin to Williams' decomposition

Comment: See also 1516

Keywords: Explicit laws, Occupation times, Enlargement of filtrations, Williams decomposition

Nature: Original

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XV: 24, 320-346, LNM 850 (1981)

**DELLACHERIE, Claude**; **LENGLART, Érik**

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (General theory of processes)

The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in 1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)

Keywords: General filtrations, Strong supermartingales, Snell's envelope, Section theorems

Nature: Original

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XVI: 22, 248-256, LNM 920 (1982)

**JEULIN, Thierry**

Sur la convergence absolue de certaines intégrales (General theory of processes)

This paper is devoted to the a.s. absolute convergence of certain random integrals, a classical example of which is $\int_0^t ds/|B_s|^{\alpha}$ for Brownian motion starting from $0$. The author does not claim to prove deep results, but his technique of optional increasing reordering (réarrangement) of a process should be useful in other contexts too

Comment: This paper greatly simplifies a proof in the author's*Semimartingales et Grossissement de Filtrations,* LNM **833**, p.44

Keywords: Enlargement of filtrations

Nature: Original

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XVI: 30, 348-354, LNM 920 (1982)

**HE, Sheng-Wu**; **WANG, Jia-Gang**

The total continuity of natural filtrations (General theory of processes)

Total continuity of a filtration ${\cal F}$ means that ${\cal F}_T={\cal F}_{T-}$ at every stopping time $T$, not necessarily previsible. It is shown that the filtration of a Lévy process without fixed discontinuities is totally continuous if and only if the jump size is a deterministic function of the jump time. Similarly, the natural filtration of a quasi-left continuous jump process is totally continuous if and only if the size of the $n$-th jump is a deterministic function of the jump times up to the $n$-th. It is shown that under the usual (here called ``strong'') previsible representation property, quasi-left continuity of the filtration implies total continuity

Keywords: Filtrations, Independent increments, Previsible representation, Total continuity, Lévy processes

Nature: Original

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XX: 31, 465-502, LNM 1204 (1986)

**McGILL, Paul**

Integral representation of martingales in the Brownian excursion filtration (Brownian motion, Stochastic calculus)

An integral representation is obtained of all square integrable martingales in the filtration $({\cal E}^x,\ x\in**R**)$, where ${\cal E}^x$ denotes the Brownian excursion $\sigma$-field below $x$ introduced by D. Williams 1343, who also showed that every $({\cal E}^x)$ martingale is continuous

Comment: Another filtration $(\tilde{\cal E}^x,\ x\in**R**)$ of Brownian excursions below $x$ has been proposed by Azéma; the structure of martingales is quite diffferent: they are discontinuous. See Y. Hu's thesis (Paris VI, 1996), and chap.~16 of Yor, *Some Aspects of Brownian Motion, Part~II*, Birkhäuser, 1997

Keywords: Previsible representation, Martingales, Filtrations

Nature: Original

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XXXII: 19, 264-305, LNM 1686 (1998)

**BARLOW, Martin T.**; **ÉMERY, Michel**; **KNIGHT, Frank B.**; **SONG, Shiqi**; **YOR, Marc**

Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)

Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA**7**, 1997). Using his methods, the result is extended to spider martingales. A conjecture of M. Barlow is also proved: if $L$ is an honest time in a (possibly multidimensional) Brownian filtration, then ${\cal F}_{L+}$ is generated by ${\cal F}_{L}$ and at most one event. Last, it is shown that a Walsh's Brownian motion can live in the filtration generated by another Walsh's Brownian motion only if the former is obtained from the latter by aggregating rays

Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,*Astérisque* **282** (2002). A simplified proof of Barlow's conjecture is given in 3304. For more on Théorème 1 (Slutsky's lemma), see 3221 and 3325

Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma

Nature: New exposition of known results, Original additions

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XXXIII: 04, 217-220, LNM 1709 (1999)

**DE MEYER, Bernard**

Une simplification de l'argument de Tsirelson sur le caractère non-brownien des processus de Walsh (Brownian motion, Filtrations)

Barlow's conjecture is proved with a simpler argument than in 3219

Keywords: Filtrations, Spider martingales

Nature: New proof of known results

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XLIII: 05, 127-186, LNM 2006 (2011)

**LAURENT, Stéphane**

On standardness and I-cosiness (Filtrations)

Keywords: Filtrations, Cosiness

Nature: Original, Exposition

On universal field equations (General theory of processes)

There is a pun in the title, since ``field'' here is a $\sigma$-field and not a quantum field. The author proves useful results on the $\sigma$-fields ${\cal F}_{T-}$ and ${\cal F}_{T+}$ associated with an arbitrary random variable $T$ in the paper of Chung-Doob,

Keywords: Filtrations

Nature: Original

Retrieve article from Numdam

XI: 22, 362-364, LNM 581 (1977)

Sur la régularisation des surmartingales (Martingale theory)

It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)

Comment: See 1524

Keywords: General filtrations, Strong supermartingales

Nature: Original

Retrieve article from Numdam

XII: 08, 57-60, LNM 649 (1978)

Sur un théorème de J. Jacod (General theory of processes)

Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals

Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration

Keywords: Semimartingales, Enlargement of filtrations, Laws of semimartingales

Nature: Original

Retrieve article from Numdam

XII: 09, 61-69, LNM 649 (1978)

Grossissement d'une filtration et semi-martingales~: théorèmes généraux (General theory of processes)

Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called

Comment: This result was independently discovered by Barlow,

Keywords: Enlargement of filtrations, Honest times

Nature: Original

Retrieve article from Numdam

XII: 10, 70-77, LNM 649 (1978)

A propos du travail de Yor sur le grossissement des tribus (General theory of processes)

This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$

Keywords: Enlargement of filtrations, Honest times

Nature: Original

Retrieve article from Numdam

XII: 11, 78-97, LNM 649 (1978)

Grossissement d'une filtration et semi-martingales~: Formules explicites (General theory of processes)

This contains very substantial improvements on 1209, namely, the explicit computation of the characteristics of the semimartingales involved

Comment: For additional results on enlargements, see the two Lecture Notes volumes

Keywords: Enlargement of filtrations, Honest times

Nature: Original

Retrieve article from Numdam

XIII: 34, 400-406, LNM 721 (1979)

Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)

This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$

Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$

Nature: Original

Retrieve article from Numdam

XIII: 50, 574-609, LNM 721 (1979)

Grossissement d'une filtration et applications (General theory of processes, Markov processes)

This is a sequel to the papers 1209 and 1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths

Keywords: Enlargement of filtrations, Williams decomposition

Nature: Original

Retrieve article from Numdam

XIV: 20, 173-188, LNM 784 (1980)

Les résultats de Jeulin sur le grossissement des tribus (General theory of processes, Stochastic calculus)

This is an introduction to beautiful results of Jeulin on enlargements, for which see

Comment: See also 1329, 1350

Keywords: Enlargement of filtrations, Semimartingales

Nature: Exposition

Retrieve article from Numdam

XIV: 21, 189-199, LNM 784 (1980)

Application d'un lemme de Jeulin au grossissement de la filtration brownienne (General theory of processes, Brownian motion)

The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment

Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')

Keywords: Enlargement of filtrations

Nature: Original

Retrieve article from Numdam

XIV: 23, 205-208, LNM 784 (1980)

Sur la compatibilité temporelle d'une tribu et d'une filtration discrète (General theory of processes)

Let us say that a $\sigma$-field ${\cal G}$ is embedded into a filtration $({\cal H}_n)$ if ${\cal G}= {\cal H}_T$ for some ${\cal H}$-stopping time $T$, that a filtration $({\cal F}_n)$ is embedded into $({\cal H}_n)$ if ${\cal F}_S$ can be embedded into $({\cal H}_n)$ for every ${\cal F}$-stopping time $S$, and finally that ${\cal G}$ and ${\cal F}$ or $({\cal F}_n)$ are compatible if they can be embedded into one single filtration $({\cal H}_n)$. Then the question investigated is whether the compatibility of ${\cal G}$ and the individual $\sigma$-fields ${\cal F}_n$ implies that of ${\cal G}$ and the whole filtration $({\cal F}_n)$

Comment: This problem arose from the spectral point of view on stochastic integration as in 1123

Keywords: Filtrations

Nature: Original

Retrieve article from Numdam

XV: 15, 210-226, LNM 850 (1981)

Sur les distributions de certaines fonctionnelles du mouvement brownien (Brownian motion)

This paper gives new proofs and extensions of results due to Knight, concerning occupation times by the process $(S_t,B_t)$ up to time $T_a$, where $(B_t)$ is Brownian motion, $T_a$ the hitting time of $a$, and $(S_t)$ is $\sup_{s\le t} B_s$. The method uses enlargement of filtrations, and martingales similar to those of 1306. Theorem 3.7 is a decomposition of Brownian paths akin to Williams' decomposition

Comment: See also 1516

Keywords: Explicit laws, Occupation times, Enlargement of filtrations, Williams decomposition

Nature: Original

Retrieve article from Numdam

XV: 24, 320-346, LNM 850 (1981)

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (General theory of processes)

The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in 1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)

Keywords: General filtrations, Strong supermartingales, Snell's envelope, Section theorems

Nature: Original

Retrieve article from Numdam

XVI: 22, 248-256, LNM 920 (1982)

Sur la convergence absolue de certaines intégrales (General theory of processes)

This paper is devoted to the a.s. absolute convergence of certain random integrals, a classical example of which is $\int_0^t ds/|B_s|^{\alpha}$ for Brownian motion starting from $0$. The author does not claim to prove deep results, but his technique of optional increasing reordering (réarrangement) of a process should be useful in other contexts too

Comment: This paper greatly simplifies a proof in the author's

Keywords: Enlargement of filtrations

Nature: Original

Retrieve article from Numdam

XVI: 30, 348-354, LNM 920 (1982)

The total continuity of natural filtrations (General theory of processes)

Total continuity of a filtration ${\cal F}$ means that ${\cal F}_T={\cal F}_{T-}$ at every stopping time $T$, not necessarily previsible. It is shown that the filtration of a Lévy process without fixed discontinuities is totally continuous if and only if the jump size is a deterministic function of the jump time. Similarly, the natural filtration of a quasi-left continuous jump process is totally continuous if and only if the size of the $n$-th jump is a deterministic function of the jump times up to the $n$-th. It is shown that under the usual (here called ``strong'') previsible representation property, quasi-left continuity of the filtration implies total continuity

Keywords: Filtrations, Independent increments, Previsible representation, Total continuity, Lévy processes

Nature: Original

Retrieve article from Numdam

XX: 31, 465-502, LNM 1204 (1986)

Integral representation of martingales in the Brownian excursion filtration (Brownian motion, Stochastic calculus)

An integral representation is obtained of all square integrable martingales in the filtration $({\cal E}^x,\ x\in

Comment: Another filtration $(\tilde{\cal E}^x,\ x\in

Keywords: Previsible representation, Martingales, Filtrations

Nature: Original

Retrieve article from Numdam

XXXII: 19, 264-305, LNM 1686 (1998)

Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)

Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA

Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,

Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma

Nature: New exposition of known results, Original additions

Retrieve article from Numdam

XXXIII: 04, 217-220, LNM 1709 (1999)

Une simplification de l'argument de Tsirelson sur le caractère non-brownien des processus de Walsh (Brownian motion, Filtrations)

Barlow's conjecture is proved with a simpler argument than in 3219

Keywords: Filtrations, Spider martingales

Nature: New proof of known results

Retrieve article from Numdam

XLIII: 05, 127-186, LNM 2006 (2011)

On standardness and I-cosiness (Filtrations)

Keywords: Filtrations, Cosiness

Nature: Original, Exposition