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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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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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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

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

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

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

Retrieve article from Numdam