Quick search | Browse volumes | |

X: 17, 245-400, LNM 511 (1976)

**MEYER, Paul-André**

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

Retrieve article from Numdam

XI: 36, 518-528, LNM 581 (1977)

**YOR, Marc**

Sur quelques approximations d'intégrales stochastiques (Martingale theory)

The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form

Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe*Publ. RIMS, Kyoto Univ.* **15**, 1979; Bismut, Mécanique Aléatoire, Springer LNM~866, 1981; Meyer 1505

Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals

Nature: Original

Retrieve article from Numdam

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

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

XLIII: 11, 269-307, LNM 2006 (2011)

**PAGÈS, Gilles**; **SELLAMI, Afef**

Convergence of multi-dimensional quantized SDE's (Integration theory, Theory of processes)

Keywords: Functional quantization, Stochastic differential equations, Stratonovich integrals, Stationary quantizers, Rough paths, Itô map, Hölder semi-norm, $p$-variation

Nature: Original

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

Retrieve article from Numdam

XI: 36, 518-528, LNM 581 (1977)

Sur quelques approximations d'intégrales stochastiques (Martingale theory)

The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form

Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe

Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals

Nature: Original

Retrieve article from Numdam

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

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

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

XLIII: 11, 269-307, LNM 2006 (2011)

Convergence of multi-dimensional quantized SDE's (Integration theory, Theory of processes)

Keywords: Functional quantization, Stochastic differential equations, Stratonovich integrals, Stationary quantizers, Rough paths, Itô map, Hölder semi-norm, $p$-variation

Nature: Original