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XV: 05, 44-102, LNM 850 (1981)

**MEYER, Paul-André**

Géométrie stochastique sans larmes (Stochastic differential geometry)

Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is*continuous semimartingales in manifolds,* following L.~Schwartz (LN **780**, 1980), but with additional features: an indication of J.M.~Bismut hinting to a definition of continuous *martingales * in a manifold, and the author's own interest on the forgotten intrinsic definition of the second differential $d^2f$ of a function. All this fits together into a geometric approach to semimartingales, and a probabilistic approach to such geometric topics as torsion-free connexions

Comment: A short introduction by the same author can be found in*Stochastic Integrals,* Springer LNM 851. The same ideas are expanded and presented in the supplement to Volume XVI and the book by Émery, *Stochastic Calculus on Manifolds *

Keywords: Semimartingales in manifolds, Martingales in manifolds, Transfer principle, Stochastic differential equations, Stochastic integrals, Stratonovich integrals

Nature: Original

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XVI-S: 59, 217-236, LNM 921 (1982)

**DARLING, Richard W.R.**

Martingales in manifolds - Definition, examples and behaviour under maps (Stochastic differential geometry)

Martingales in manifolds have been introduced independently by Meyer 1505 and the author (Ph.D. Thesis). This short note is a review of that thesis; here, the definition of a manifold-valued martingale is by its behaviour under convex functions

Comment: More details are given in*Bull. L.M.S.* **15** (1983), *Publ R.I.M.S. Kyoto*~**19** (1983) and *Zeit. für W-theorie* **65** (1984). Characterizating of manifold-valued martingales by convex functions has become a powerful tool: see for instance Émery's book *Stochastic Calculus in Manifolds* (Springer, 1989) and his St-Flour lectures (Springer LNM 1738)

Keywords: Martingales in manifolds, Semimartingales in manifolds, Convex functions

Nature: Original

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XVII: 18, 179-184, LNM 986 (1983)

**HE, Sheng-Wu**; **YAN, Jia-An**; **ZHENG, Wei-An**

Sur la convergence des semimartingales continues dans ${\bf R}^n$ et des martingales dans une variété (Stochastic calculus, Stochastic differential geometry)

Say that a continuous semimartingale $X$ with canonical decomposition $X_0+M+A$ converges perfectly on an event $E$ if both $M_t$ and $\int_0^t|dA_s|$ have an a.s. limit on $E$ when $t\rightarrow \infty $. It is established that if $A_t$ has the form $\int_0^tH_sd[M,M]_s$, $X$ converges perfectly on the event $\{\sup_t|X_t|+\lim\sup_tH_t <\infty \}$. A similar (but less simple) statement is shown for multidimensional $X$; and an application is given to martingales in manifolds: every point of a manifold $V$ (with a connection) has a neighbourhood $U$ such that, given any $V$-valued martingale $X$, almost all paths of $X$ that eventually remain in $U$ are convergent

Comment: The latter statement (martingale convergence) is very useful; more recent proofs use convex functions instead of perfect convergence. The next talk 1719 is a small remark on perfect convergence

Keywords: Semimartingales, Martingales in manifolds

Nature: Original

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XVII: 20, 187-193, LNM 986 (1983)

**MEYER, Paul-André**

Le théorème de convergence des martingales dans les variétés riemanniennes, d'après R.W. Darling et W.A. Zheng (Stochastic differential geometry)

Exposition of two results on the asymptotic behaviour of martingales in a Riemannian manifold: First, Darling's theorem says that on the event where the Riemannian quadratic variation $<X,X>_\infty$ of a martingale $X$ is finite, $X_\infty$ exists in the Aleksandrov compactification of $V$. Second, Zheng's theorem asserts that on the event where $X_\infty$ exists in $V$, the Riemannian quadratic variation $<X,X>_\infty$ is finite

Comment: Darling's result is in*Publ. R.I.M.S. Kyoto* **19** (1983) and Zheng's in *Zeit. für W-theorie* **63** (1983). As observed in He-Yan-Zheng 1718, a stronger version of Zheng's theorem holds (with the same argument): On the event where $X_\infty$ exists in $V$, $X$ is a semimartingale up to infinity (so for instance solutions to good SDE's driven by $X$ also have a limit at infinity)

Keywords: Martingales in manifolds

Nature: Exposition

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XVIII: 33, 501-518, LNM 1059 (1984)

**ÉMERY, Michel**; **ZHENG, Wei-An**

Fonctions convexes et semimartingales dans une variété (Stochastic differential geometry)

On a manifold endowed with a connexion, convex functions can be defined, and transform manifold-valued martingales into real-valued local submartingales (see Darling 1659). This is extended here to the case of non-smooth convex functions. Ii is also shown that they make manifold-valued semimartingales into real semimartingales

Keywords: Semimartingales in manifolds, Martingales in manifolds, Convex functions

Nature: Original

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XX: 23, 352-374, LNM 1204 (1986)

**HAKIM-DOWEK, M.**; **LÉPINGLE, Dominique**

L'exponentielle stochastique des groupes de Lie (Stochastic differential geometry)

Given a Lie group $G$ and its Lie algebra $\cal G$, this article defines and studies the stochastic exponential of a (continuous) semimartingale $M$ in $\cal G$ as the solution in $G$ to the Stratonovich s.d.e. $dX = X dM$. The inverse operation (stochastic logarithm) is also considered; various formulas are established (e.g. the exponential of $M+N$). When $M$ is a local martingale, $X$ is a martingale for the connection such that $\nabla_A B=0$ for all left-invariant vector fields $A$ and $B$

Comment: See also Karandikar*Ann. Prob.* **10** (1982) and 1722. For a sequel, see Arnaudon 2612

Keywords: Semimartingales in manifolds, Martingales in manifolds, Lie group

Nature: Original

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XXV: 18, 196-219, LNM 1485 (1991)

**PICARD, Jean**

Calcul stochastique avec sauts sur une variété (Stochastic differential geometry)

It is known from Meyer 1505 that intrinsic Ito integrals have a meaning for continuous semimartingales in a manifold $M$, provided $M$ is endowed with a connection. This is extended here to càdlàg semimartingales. The manifold must be endowed with a richer structure, a ``connector'', mapping $M\times M$ to the tangent bundle, that allows to interpret a jump $(X_{t-},X_t)$ as a tangent vector to $M$ at $X{t-}$; the differential of the connector at the diagonal reduces to a classical torsion-free connection. Introducing torsions leads to a more general ``transporter'', describing how parallel transports should behave at jump times, and reducing to a classical connection for infinitesimal jumps. Discrete-time approximations are established.

Keywords: Semimartingales in manifolds, Martingales in manifolds, Jumps

Nature: Original

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XXV: 19, 220-233, LNM 1485 (1991)

**ÉMERY, Michel**; **MOKOBODZKI, Gabriel**

Sur le barycentre d'une probabilité dans une variété (Stochastic differential geometry)

In a manifold $V$ (endowed with a connection), convex functions and continuous martingales can be defined, but expectations cannot. This article proposes to define the mass-centre of a probability $\mu$ on $V$ as a whole set of points, consisting of all $x$ in $V$ such that $f(x)\le\mu(f)$ for all bounded, convex $f$ on $V$. If $V$ is small enough, it is shown that this is equivalent to demanding that there exists (on a suitable filtered probability space) a continuous martingale $X$ such that $X_0=x$ and $X_1$ has law $\mu$

Comment: The conjecture (due to Émery) at the bottom of page 232 has been disproved by Kendall (*J. London Math. Soc.* **46**, 1992), as pointed out in 2650

Keywords: Martingales in manifolds, Convex functions

Nature: Original

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XXVI: 12, 146-154, LNM 1526 (1992)

**ARNAUDON, Marc**

Connexions et martingales dans les groupes de Lie (Stochastic differential geometry)

The stochastic exponential of Hakim-Dowek-Lépingle 2023 is interpreted in terms of second-order geometry, studied in details and generalized

Keywords: Martingales in manifolds, Lie group

Nature: Original

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XXVI: 13, 155-156, LNM 1526 (1992)

**ARNAUDON, Marc**; **MATTHIEU, Pierre**

Appendice : Décomposition en produit de deux browniens d'une martingale à valeurs dans un groupe muni d'une métrique bi-invariante (Stochastic differential geometry)

Using 2612, it is shown that in a Lie group with a bi-invariant Riemannian structure, every martingale is a time-changed product of two Brownian motions

Keywords: Martingales in manifolds, Lie group

Nature: Original

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XXVI: 50, 633-633, LNM 1526 (1992)

**ÉMERY, Michel**; **MOKOBODZKI, Gabriel**

Correction au Séminaire~XXV (Stochastic differential geometry)

Points out that the conjecture (due to Émery) at the bottom of page 232 in 2519 is refuted by Kendall (*J. London Math. Soc.* **46**, 1992)

Keywords: Martingales in manifolds

Nature: Correction

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Géométrie stochastique sans larmes (Stochastic differential geometry)

Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is

Comment: A short introduction by the same author can be found in

Keywords: Semimartingales in manifolds, Martingales in manifolds, Transfer principle, Stochastic differential equations, Stochastic integrals, Stratonovich integrals

Nature: Original

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XVI-S: 59, 217-236, LNM 921 (1982)

Martingales in manifolds - Definition, examples and behaviour under maps (Stochastic differential geometry)

Martingales in manifolds have been introduced independently by Meyer 1505 and the author (Ph.D. Thesis). This short note is a review of that thesis; here, the definition of a manifold-valued martingale is by its behaviour under convex functions

Comment: More details are given in

Keywords: Martingales in manifolds, Semimartingales in manifolds, Convex functions

Nature: Original

Retrieve article from Numdam

XVII: 18, 179-184, LNM 986 (1983)

Sur la convergence des semimartingales continues dans ${\bf R}^n$ et des martingales dans une variété (Stochastic calculus, Stochastic differential geometry)

Say that a continuous semimartingale $X$ with canonical decomposition $X_0+M+A$ converges perfectly on an event $E$ if both $M_t$ and $\int_0^t|dA_s|$ have an a.s. limit on $E$ when $t\rightarrow \infty $. It is established that if $A_t$ has the form $\int_0^tH_sd[M,M]_s$, $X$ converges perfectly on the event $\{\sup_t|X_t|+\lim\sup_tH_t <\infty \}$. A similar (but less simple) statement is shown for multidimensional $X$; and an application is given to martingales in manifolds: every point of a manifold $V$ (with a connection) has a neighbourhood $U$ such that, given any $V$-valued martingale $X$, almost all paths of $X$ that eventually remain in $U$ are convergent

Comment: The latter statement (martingale convergence) is very useful; more recent proofs use convex functions instead of perfect convergence. The next talk 1719 is a small remark on perfect convergence

Keywords: Semimartingales, Martingales in manifolds

Nature: Original

Retrieve article from Numdam

XVII: 20, 187-193, LNM 986 (1983)

Le théorème de convergence des martingales dans les variétés riemanniennes, d'après R.W. Darling et W.A. Zheng (Stochastic differential geometry)

Exposition of two results on the asymptotic behaviour of martingales in a Riemannian manifold: First, Darling's theorem says that on the event where the Riemannian quadratic variation $<X,X>_\infty$ of a martingale $X$ is finite, $X_\infty$ exists in the Aleksandrov compactification of $V$. Second, Zheng's theorem asserts that on the event where $X_\infty$ exists in $V$, the Riemannian quadratic variation $<X,X>_\infty$ is finite

Comment: Darling's result is in

Keywords: Martingales in manifolds

Nature: Exposition

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XVIII: 33, 501-518, LNM 1059 (1984)

Fonctions convexes et semimartingales dans une variété (Stochastic differential geometry)

On a manifold endowed with a connexion, convex functions can be defined, and transform manifold-valued martingales into real-valued local submartingales (see Darling 1659). This is extended here to the case of non-smooth convex functions. Ii is also shown that they make manifold-valued semimartingales into real semimartingales

Keywords: Semimartingales in manifolds, Martingales in manifolds, Convex functions

Nature: Original

Retrieve article from Numdam

XX: 23, 352-374, LNM 1204 (1986)

L'exponentielle stochastique des groupes de Lie (Stochastic differential geometry)

Given a Lie group $G$ and its Lie algebra $\cal G$, this article defines and studies the stochastic exponential of a (continuous) semimartingale $M$ in $\cal G$ as the solution in $G$ to the Stratonovich s.d.e. $dX = X dM$. The inverse operation (stochastic logarithm) is also considered; various formulas are established (e.g. the exponential of $M+N$). When $M$ is a local martingale, $X$ is a martingale for the connection such that $\nabla_A B=0$ for all left-invariant vector fields $A$ and $B$

Comment: See also Karandikar

Keywords: Semimartingales in manifolds, Martingales in manifolds, Lie group

Nature: Original

Retrieve article from Numdam

XXV: 18, 196-219, LNM 1485 (1991)

Calcul stochastique avec sauts sur une variété (Stochastic differential geometry)

It is known from Meyer 1505 that intrinsic Ito integrals have a meaning for continuous semimartingales in a manifold $M$, provided $M$ is endowed with a connection. This is extended here to càdlàg semimartingales. The manifold must be endowed with a richer structure, a ``connector'', mapping $M\times M$ to the tangent bundle, that allows to interpret a jump $(X_{t-},X_t)$ as a tangent vector to $M$ at $X{t-}$; the differential of the connector at the diagonal reduces to a classical torsion-free connection. Introducing torsions leads to a more general ``transporter'', describing how parallel transports should behave at jump times, and reducing to a classical connection for infinitesimal jumps. Discrete-time approximations are established.

Keywords: Semimartingales in manifolds, Martingales in manifolds, Jumps

Nature: Original

Retrieve article from Numdam

XXV: 19, 220-233, LNM 1485 (1991)

Sur le barycentre d'une probabilité dans une variété (Stochastic differential geometry)

In a manifold $V$ (endowed with a connection), convex functions and continuous martingales can be defined, but expectations cannot. This article proposes to define the mass-centre of a probability $\mu$ on $V$ as a whole set of points, consisting of all $x$ in $V$ such that $f(x)\le\mu(f)$ for all bounded, convex $f$ on $V$. If $V$ is small enough, it is shown that this is equivalent to demanding that there exists (on a suitable filtered probability space) a continuous martingale $X$ such that $X_0=x$ and $X_1$ has law $\mu$

Comment: The conjecture (due to Émery) at the bottom of page 232 has been disproved by Kendall (

Keywords: Martingales in manifolds, Convex functions

Nature: Original

Retrieve article from Numdam

XXVI: 12, 146-154, LNM 1526 (1992)

Connexions et martingales dans les groupes de Lie (Stochastic differential geometry)

The stochastic exponential of Hakim-Dowek-Lépingle 2023 is interpreted in terms of second-order geometry, studied in details and generalized

Keywords: Martingales in manifolds, Lie group

Nature: Original

Retrieve article from Numdam

XXVI: 13, 155-156, LNM 1526 (1992)

Appendice : Décomposition en produit de deux browniens d'une martingale à valeurs dans un groupe muni d'une métrique bi-invariante (Stochastic differential geometry)

Using 2612, it is shown that in a Lie group with a bi-invariant Riemannian structure, every martingale is a time-changed product of two Brownian motions

Keywords: Martingales in manifolds, Lie group

Nature: Original

Retrieve article from Numdam

XXVI: 50, 633-633, LNM 1526 (1992)

Correction au Séminaire~XXV (Stochastic differential geometry)

Points out that the conjecture (due to Émery) at the bottom of page 232 in 2519 is refuted by Kendall (

Keywords: Martingales in manifolds

Nature: Correction

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