nach oben

1989 | Buch

Kapitel lesen Erstes Kapitel lesen

Stochastic Calculus in Manifolds

verfasst von: Michel Emery

Verlag: Springer Berlin Heidelberg

Buchreihe : Universitext

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

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

Frontmatter

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 (ℱ_t)_{t ≥0}: each ℱ_tis a sub-σ-field of ℱ, contains all negligible events in ℱand ℱ_t=∩_ε>0 ℱ_t+ε; this equality says that t↦ℱ_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,(ℱ_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 x∊M, a Euclidean structure is defined on the vector space T_xM 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 Σ_λ f^λ dh^λ and defining the integral as I = Σ_λ ∫f^λo X d(hλoX); but this fails by lack of intrinsicness. Indeed, it amounts to transforming α into the second order form θ = Σ_λ f^λ d² h^λ 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) — udv — vdu 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

Titel: Stochastic Calculus in Manifolds
verfasst von: Michel Emery
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-75051-9
Print ISBN: 978-3-540-51664-4
DOI: https://doi.org/10.1007/978-3-642-75051-9

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter I. Real semimartingales and stochastic integrals

Chapter II. Some vocabulary from differential geometry

Chapter III. Manifold-valued semimartingales and their quadratic variation

Chapter IV. Connections and martingales

Chapter V. Riemannian manifolds and Brownian motions

Chapter VI. Second order vectors and forms

Chapter VII. Stranovich and Itô integrals of first order forms

Chapter VIII. Parallel transport and moving frame

Backmatter