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## Über dieses Buch

Addressed to both pure and applied probabilitists, including graduate students, this text is a pedagogically-oriented introduction to the Schwartz-Meyer second-order geometry and its use in stochastic calculus. P.A. Meyer has contributed an appendix: "A short presentation of stochastic calculus" presenting the basis of stochastic calculus and thus making the book better accessible to non-probabilitists also. No prior knowledge of differential geometry is assumed of the reader: this is covered within the text to the extent. The general theory is presented only towards the end of the book, after the reader has been exposed to two particular instances - martingales and Brownian motions - in manifolds. The book also includes new material on non-confluence of martingales, s.d.e. from one manifold to another, approximation results for martingales, solutions to Stratonovich differential equations. Thus this book will prove very useful to specialists and non-specialists alike, as a self-contained introductory text or as a compact reference.

## Inhaltsverzeichnis

### Chapter I. Real semimartingales and stochastic integrals

Abstract
The usual setup for the general theory of processes is a complete probability space (Ω,ℱ,ℙ) endowed with a filtration ( t )t ≥0: each tis a sub-σ-field of , contains all negligible events in and t =∩ε>0 t+ε ; this equality says that t t is increasing and right-continuous. Increasingness is the main point; the other conditions are mere technical assumptions. Removing them is possible, but rarely useful, and makes some results heavier.
Michel Emery

### Chapter II. Some vocabulary from differential geometry

Abstract
This chapter is almost empty; it consists mainly of definitions that can be found at the beginning of any textbook on elementary differential geometry (the only non-trivial result, Whitney’s imbedding theorem, is admitted). We urge the reader who has not forgotten every single word of his or her undergraduate course in differential geometry to skip it, remembering only that, in these notes, ‘form’ stands for differential form of degree 1 and ‘bilinear form’ for tensor of rank (0,2) (i.e. twice covariant).
Michel Emery

### Chapter III. Manifold-valued semimartingales and their quadratic variation

Abstract
Let M denote a manifold. A continuous M-valued process X, defined on some filtered probability space (Ω,ℱ,ℙ,( t )t≥0) is called a semimartingale if, for each smooth function f on M, the real-valued process f o X is a semimartingale. When M is the real line or ℝn (with the canonical C structure!), this definition agrees with the usual one.
Michel Emery

### Chapter IV. Connections and martingales

Abstract
The main point in this chapter is the possibility, on some manifolds, of distinguishing among semimartingales a smaller class, that of martingales. The name ‘local martingale’ would fit these processes better, since when the manifold is the flat space ℝ or ℝn they are exactly the usual (continuous) local martingales; but we won’t consider any subclass corresponding in that case to usual martingales, so no ambiguity is created by calling them just martingales (except precisely in that flat case!).
Michel Emery

### Chapter V. Riemannian manifolds and Brownian motions

Abstract
A Riemannian manifold is a pair (M,g) where g is a symmetric, positive definite bilinear form on M, called the metric tensor. At each point xM, a Euclidean structure is defined on the vector space T x M by the bilinear form g(x), making its dual T x * M canonically isomorphic to it.
Michel Emery

### Chapter VI. Second order vectors and forms

Abstract
Given a manifold M, denote by E a C-module of functions on M containing C and such that belonging to E is a local property (for instance all Borel, or locally bounded, or continuous, or C p or C functions). Let L be a linear mapping from C to E (but not necessarily C-linear), and denote by Γthe “squared field operator” associated to L:
$$\Gamma (f,g) = \frac{1}{2}[L(fg) - fLg - gLf].$$
.
Michel Emery

### Chapter VII. Stranovich and Itô integrals of first order forms

Abstract
If α is a form (of order 1) on M, can one integrate α along a semimartingale X? A possible attempt, inspired by (6.24) and (6.25) could consist in writing α as a finite sum Σλ fλ dhλ and defining the integral as I = Σλ ∫fλo X d(hλoX); but this fails by lack of intrinsicness. Indeed, it amounts to transforming α into the second order form θ = Σλ fλ d2 hλ and letting I = θ,dX〉; but θ so obtained does depend upon the decomposition of α. If for instance u and v are smooth functions, the form α = d(uv) — udvvdu vanishes identically, but θ constructed as above is 2du.dv, and I = [uoX, voX] is not null in general.
Michel Emery

### Chapter VIII. Parallel transport and moving frame

Abstract
Consider a smooth curve A in the tangent manifold TM, that is to say a smooth curve γ in M, and for each t, a tangent vector A (t) to M at γ(t) that depends smoothly upon t; such an object is called a transport (if you imagine t as the time, γ is a moving point in M and A can be seen as a vector carried — transported — along the curve by this moving point). As is customary in geometry, we shall denote by π the smooth mapping from TM to M defined by $$\pi a = x \Leftrightarrow a \in {T_x}M,\;so\;\gamma \;is\;just\;\pi \;A.$$
Michel Emery

### Backmatter

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