Quick search | Browse volumes | |

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

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

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

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

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

Retrieve article from Numdam

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

Retrieve article from Numdam