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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: 57, 165-207, LNM 921 (1982)

**MEYER, Paul-André**

Géométrie différentielle stochastique (bis) (Stochastic differential geometry)

A sequel to 1505. The main theme is that an ordinary differential equation has a non unique extension as a stochastic differential equation: besides the Stratonovich one, given by the ``transfer principle'', there are other possibilities: choosing among them requires some additional, connection-like, structure. The most striking application is the Dohrn-Guerra correction to the parallel transport along a semimartingale

Comment: For complements, see Émery 1658, Hakim-Dowek-Lépingle 2023, Émery's monography*Stochastic Calculus in Manifolds* (Springer, 1989) and article 2428, and Arnaudon-Thalmaier 3214

Keywords: Semimartingales in manifolds, Stochastic differential equations, Local characteristics, Nelson's stochastic mechanics, Transfer principle

Nature: Original

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XVI-S: 58, 208-216, LNM 921 (1982)

**ÉMERY, Michel**

En marge de l'exposé de Meyer : ``Géométrie différentielle stochastique'' (Stochastic differential geometry)

Marginal remarks to Meyer 1657

Keywords: Semimartingales in manifolds, Stochastic differential equations

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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XVII: 21, 194-197, LNM 986 (1983)

**PRICE, Gareth C.**; **WILLIAMS, David**

Rolling with `slipping': I (Stochastic calculus, Stochastic differential geometry)

If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$

Keywords: Brownian motion in a manifold, Previsible representation

Nature: Original

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XVII: 22, 198-204, LNM 986 (1983)

**KARANDIKAR, Rajeeva L.**

Girsanov type formula for a Lie group valued Brownian motion (Brownian motion, Stochastic differential geometry)

A formula for the change of measure of a Lie group valued Brownian motion is stated and proved. It needs a Borel correspondence between paths in the Lie algebra and paths in the group, that transforms all (continuous) semimartingales in the algebra into their stochastic exponential

Comment: For more on stochastic exponentials in Lie groups, see Hakim-Dowek-Lépingle 2023 and Arnaudon 2612

Keywords: Changes of measure, Brownian motion in a manifold, Lie group

Nature: Original

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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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XIX: 07, 91-112, LNM 1123 (1985)

**SCHWARTZ, Laurent**

Construction directe d'une diffusion sur une variété (Stochastic differential geometry)

This seems to be the first use of Witney's embedding theorem to construct a process (a Brownian motion, a diffusion, a solution to some s.d.e.) in a manifold $M$ by embedding $M$ into some $**R**^d$. Very general existence and uniqueness results are obtained

Comment: This method has since become standard in stochastic differential geometry; see for instance Émery's book*Stochastic Calculus in Manifolds* (Springer, 1989)

Keywords: Diffusions in manifolds, Stochastic differential equations

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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XXIV: 28, 407-441, LNM 1426 (1990)

**ÉMERY, Michel**

On two transfer principles in stochastic differential geometry (Stochastic differential geometry)

Second-order stochastic calculus gives two intrinsic methods to transform an ordinary differential equation into a stochastic one (see Meyer 1657, Schwartz 1655 or Emery*Stochastic calculus in manifolds*). The first one gives a Stratonovich SDE and needs coefficients regular enough; the second one gives an Ito equation and needs a connection on the manifold. Discretizing time and smoothly interpolating the driving semimartingale is known to give an approximation to the Stratonovich transfer; it is shown here that another discretized-time procedure converges to the Ito transfer. As an application, if the ODE makes geodesics to geodesics, then the Ito and Stratonovich SDE's are the same

Comment: An error is corrected in 2649. The term ``transfer principle'' was coined by Malliavin,*Géométrie Différentielle Stochastique,* Presses de l'Université de Montréal (1978); see also Bismut, *Principes de Mécanique Aléatoire* (1981) and 1505

Keywords: Stochastic differential equations, Semimartingales in manifolds, Transfer principle

Nature: Original

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XXIV: 30, 448-452, LNM 1426 (1990)

**ÉMERY, Michel**; **LÉANDRE, Rémi**

Sur une formule de Bismut (Markov processes, Stochastic differential geometry)

This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group

Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes

Nature: Exposition, Original additions

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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: 10, 113-126, LNM 1526 (1992)

**TAYLOR, John C.**

Skew products, regular conditional probabilities and stochastic differential equations: a technical remark (Stochastic calculus, Stochastic differential geometry)

This is a detailed study of the transfer principle (the solution to a Stratonovich stochastic differential equations can be pathwise obtained from the driving semimartingale by solving the corresponding ordinary differential equation) in the case of an equation where the solution of another equation plays the role of a parameter

Comment: The term ``transfer principle'' was coined by Malliavin,*Géométrie Différentielle Stochastique,* Presses de l'Université de Montréal (1978); see also Bismut, *Principes de Mécanique Aléatoire* (1981)

Keywords: Transfer principle, Stochastic differential equations, Stratonovich integrals

Nature: Original

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XXVI: 11, 127-145, LNM 1526 (1992)

**ESTRADE, Anne**; **PONTIER, Monique**

Relèvement horizontal d'une semimartingale càdlàg (Stochastic differential geometry, Stochastic calculus)

For filtering purposes, the lifting of a manifold-valued semimartingale $X$ to the tangent space at $X_0$ is extended here to the case when $X$ has jumps. The value of $L_t$ involves the inverse of the exponential at $X_{t-}$ applied to $X_t$, and a parallel transport from $X_0$ to $X_{t-}$

Comment: The same method is described in a more general setting by Kurtz-Pardoux-Protter*Ann I.H.P.* (1995). In turn, this is a particular instance of a very general scheme due to Cohen (*Stochastics Stoch. Rep.* (1996)

Keywords: Stochastic parallel transport, Stochastic differential equations, Jumps

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: 18, 189-209, LNM 1526 (1992)

**NORRIS, James R.**

A complete differential formalism for stochastic calculus in manifolds (Stochastic differential geometry)

The use of equivariant coordinates in stochastic differential geometry is replaced here by an equivalent, but intrinsic, formalism, where the differential of a semimartingale lives in the tangent bundle. Simple, intrinsic Girsanov and Feynman-Kac formulas are given, as well as a nice construction of a Brownian motion in a manifold admitting a Riemannian submersion with totally geodesic fibres

Keywords: Semimartingales in manifolds, Stochastic integrals, Feynman-Kac formula, Changes of measure, Heat semigroup

Nature: Original

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XXVI: 49, 633-633, LNM 1526 (1992)

**ÉMERY, Michel**

Correction au Séminaire~XXIV (Stochastic differential geometry)

An error in 2428 is pointed out; it is corrected by Cohen (*Stochastics Stochastics Rep.* **56**, 1996)

Keywords: Stochastic differential equations, Semimartingales in manifolds

Nature: Correction

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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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XXIX: 16, 166-180, LNM 1613 (1995)

**APPLEBAUM, David**

A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)

This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.

Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (*Stochastics Stochastics Rep.* **56**, 1996). The same question is addressed by Cohen in the next article 2917

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

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XXIX: 17, 181-193, LNM 1613 (1995)

**COHEN, Serge**

Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)

The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (*Stochastics Stochastics Rep.* **56**, 1996). Here this language is used to study the Markov property of solutions to SDE's with jumps. In particular,two definitions of a Lévy process in a Riemannian manifold are compared: One as the solution to a SDE driven by some Euclidean Lévy process, the other by subordinating some Riemannian Brownian motion. It is shown that in general the former is not of the second kind

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

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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: 57, 165-207, LNM 921 (1982)

Géométrie différentielle stochastique (bis) (Stochastic differential geometry)

A sequel to 1505. The main theme is that an ordinary differential equation has a non unique extension as a stochastic differential equation: besides the Stratonovich one, given by the ``transfer principle'', there are other possibilities: choosing among them requires some additional, connection-like, structure. The most striking application is the Dohrn-Guerra correction to the parallel transport along a semimartingale

Comment: For complements, see Émery 1658, Hakim-Dowek-Lépingle 2023, Émery's monography

Keywords: Semimartingales in manifolds, Stochastic differential equations, Local characteristics, Nelson's stochastic mechanics, Transfer principle

Nature: Original

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XVI-S: 58, 208-216, LNM 921 (1982)

En marge de l'exposé de Meyer : ``Géométrie différentielle stochastique'' (Stochastic differential geometry)

Marginal remarks to Meyer 1657

Keywords: Semimartingales in manifolds, Stochastic differential equations

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

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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

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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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XVII: 21, 194-197, LNM 986 (1983)

Rolling with `slipping': I (Stochastic calculus, Stochastic differential geometry)

If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$

Keywords: Brownian motion in a manifold, Previsible representation

Nature: Original

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XVII: 22, 198-204, LNM 986 (1983)

Girsanov type formula for a Lie group valued Brownian motion (Brownian motion, Stochastic differential geometry)

A formula for the change of measure of a Lie group valued Brownian motion is stated and proved. It needs a Borel correspondence between paths in the Lie algebra and paths in the group, that transforms all (continuous) semimartingales in the algebra into their stochastic exponential

Comment: For more on stochastic exponentials in Lie groups, see Hakim-Dowek-Lépingle 2023 and Arnaudon 2612

Keywords: Changes of measure, Brownian motion in a manifold, Lie group

Nature: Original

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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

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XIX: 07, 91-112, LNM 1123 (1985)

Construction directe d'une diffusion sur une variété (Stochastic differential geometry)

This seems to be the first use of Witney's embedding theorem to construct a process (a Brownian motion, a diffusion, a solution to some s.d.e.) in a manifold $M$ by embedding $M$ into some $

Comment: This method has since become standard in stochastic differential geometry; see for instance Émery's book

Keywords: Diffusions in manifolds, Stochastic differential equations

Nature: Original

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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

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XXIV: 28, 407-441, LNM 1426 (1990)

On two transfer principles in stochastic differential geometry (Stochastic differential geometry)

Second-order stochastic calculus gives two intrinsic methods to transform an ordinary differential equation into a stochastic one (see Meyer 1657, Schwartz 1655 or Emery

Comment: An error is corrected in 2649. The term ``transfer principle'' was coined by Malliavin,

Keywords: Stochastic differential equations, Semimartingales in manifolds, Transfer principle

Nature: Original

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XXIV: 30, 448-452, LNM 1426 (1990)

Sur une formule de Bismut (Markov processes, Stochastic differential geometry)

This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group

Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes

Nature: Exposition, Original additions

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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

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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

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XXVI: 10, 113-126, LNM 1526 (1992)

Skew products, regular conditional probabilities and stochastic differential equations: a technical remark (Stochastic calculus, Stochastic differential geometry)

This is a detailed study of the transfer principle (the solution to a Stratonovich stochastic differential equations can be pathwise obtained from the driving semimartingale by solving the corresponding ordinary differential equation) in the case of an equation where the solution of another equation plays the role of a parameter

Comment: The term ``transfer principle'' was coined by Malliavin,

Keywords: Transfer principle, Stochastic differential equations, Stratonovich integrals

Nature: Original

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XXVI: 11, 127-145, LNM 1526 (1992)

Relèvement horizontal d'une semimartingale càdlàg (Stochastic differential geometry, Stochastic calculus)

For filtering purposes, the lifting of a manifold-valued semimartingale $X$ to the tangent space at $X_0$ is extended here to the case when $X$ has jumps. The value of $L_t$ involves the inverse of the exponential at $X_{t-}$ applied to $X_t$, and a parallel transport from $X_0$ to $X_{t-}$

Comment: The same method is described in a more general setting by Kurtz-Pardoux-Protter

Keywords: Stochastic parallel transport, Stochastic differential equations, Jumps

Nature: Original

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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

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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

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XXVI: 18, 189-209, LNM 1526 (1992)

A complete differential formalism for stochastic calculus in manifolds (Stochastic differential geometry)

The use of equivariant coordinates in stochastic differential geometry is replaced here by an equivalent, but intrinsic, formalism, where the differential of a semimartingale lives in the tangent bundle. Simple, intrinsic Girsanov and Feynman-Kac formulas are given, as well as a nice construction of a Brownian motion in a manifold admitting a Riemannian submersion with totally geodesic fibres

Keywords: Semimartingales in manifolds, Stochastic integrals, Feynman-Kac formula, Changes of measure, Heat semigroup

Nature: Original

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XXVI: 49, 633-633, LNM 1526 (1992)

Correction au Séminaire~XXIV (Stochastic differential geometry)

An error in 2428 is pointed out; it is corrected by Cohen (

Keywords: Stochastic differential equations, Semimartingales in manifolds

Nature: Correction

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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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XXIX: 16, 166-180, LNM 1613 (1995)

A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)

This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.

Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

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XXIX: 17, 181-193, LNM 1613 (1995)

Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)

The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

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