With an Appendix by P.A. Meyer
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.
I. Real semimartingales and stochastic integrals.- 1.1 Filtration, Process, Predictable.- 1.2 Stopping time, Stochastic interval, Stopped process.- 1.3 Convergence in probability uniformly on compact sets, Subdivision, Size of a subdivision.- 1.4 Change of time.- 1.5 Martingale, Continuous local martingale, Process with finite variation. Semimartingale, Local submartingale, Semimartingale up to infinity.- 1.6 Locally bounded, Stochastic integral.- 1.7 Quadratic variation of semimartingales.- 1.10 Change of variable formula.- 1.12 Stratonovich integral.- 1.16, 17 Existence, uniqueness and stability for the solution to a stochastic differential equation.- II. Some vocabulary from differential geometry.- 2.1 Manifold.- 2.2 Whitney's imbedding theorem.- 2.3 Tangent vector, Tangent space.- 2.6 Push-forward of a vector.- 2.7 Speed of a curve.- 2.10 Tangent manifold, Vector field.- 2.14 Cotangent vector, Covector, Form at a given point.- 2.15 Form.- 2.18 Pull-back of a form.- 2.20 Bilinear form.- 2.24 Pull-back of a bilinear form.- 2.25 Flow of a vector field, Lie-derivative of a function.- 2.26 Lie-derivative of a vector field, Commutator of two vector fields.- 2.30 Lie-derivative of a form.- 2.32 Lie-derivative of a bilinear form.- III. Manifold-valued semimartingales and their quadratic variation.- 3.1 M-valued semimartingale.- 3.4 Localness of M-valued semimartingales.- 3.5 Space-localness implies time-localness.- 3.7 M-valued semimartingale in an interval, M-valued semimartingale up to infinity.- 3.8, 9 b-quadratic variation of a semimartingale.- 3.13 Change of space in a b-quadratic variation.- 3.23 Discrete approximation of ? b(dX,dX).- IV. Connections and martingales.- 4.1 Connection.- 4.2 Martingale.- 4.6 Localization of martingales.- 4.7 Martingale on an interval.- 4.8 Flat connection.- 4.9 Induced connection on a submanifold of ?N.- 4.10 Martingales for this connection.- 4.13 Change of variable formula for IIess.- 4.16 Christoffel symbols.- 4.17 Expression of a connection in local coordinates.- 4.18 Change of chart formula for Christoffel symbols.- 4.19 Equation of martingales.- 4.21 Affine function.- 4.25 Geodesic.- 4.27 Equation of geodesics.- 4.31 Geodesic in a submanifold of ?N.- 4.32 Characterization of affine functions by geodesics or martingales.- 4.33 Characterization of connections by geodesics or martingales.- 4.35 Convex function.- 4.37 Characterization of convex functions with geodesics or martingales.- 4.39, 41 Characterization of geodesics and martingales with convex functions.- 4.43 A uniform limit of martingales is a martingale.- 4.46 Convergence of martingales in a small manifold.- 4.48 A convergent martingale is a semimartingale up to infinity.- 4.52 Totally geodesic submanifold.- 4.58 Product connection.- 4.61 Non-confluence of martingales.- V. Riemannian manifolds and Brownian motions.- 5.1 Riemannian manifold.- 5.2 Gradient, length, energy, ??dX|dX?.- 5.4 Riemannian submanifold.- 5.5 Canonical (Levi-Civita) connection.- 5.8 Variational characterization of geodesics.- 5.12 Riemannian expression of the Christoffel symbols.- 5.14 Laplacian.- 5.16 Brownian motion.- 5.18 Characterization of Brownian motions.- 5.23 Change of variable formula for the Laplacian.- 5.24 Harmonic mapping.- 5.29 Geodesic-completeness.- 5.32 Darling-Zheng convergence theorem for Riemannian martingales.- 5.34 Martingale-completeness.- 5.35 Brownian-completeness.- 5.37 Sufficent condition for completeness.- 5.39-43 Examples of completeness and non-completeness.- VI. Second order vectors and forms.- 6.1 Equivalent definitions of second order vectors.- 6.3 Tangent vectors of order 2.- 6.5 Acceleration of a curve.- 6.7 Push-forward of second order vectors.- 6.8 Vector of order 2 written in local coordinates.- 6.10 Forms of order 2.- 6.11 Product ?.? of two first order forms.- 6.12 Restriction to order 1.- 6.13 Second order form associated to a bilinear form.- 6.15 Form of order 2 written in local coordinates.- 6.19 Pull-back of a form of order 2.- 6.21 Schwartz' principle, dX.- 6.22 Schwartz morphism.- 6.24-31 Integration of second order forms against semimartingales.- 6.33 Intrinsic stochastic differential equation in manifolds.- 6.34 Schwartz operator.- 6.35 Stochastic differential equation dY = f (X,Y)dX.- 6.41 Existence and uniqueness for the solution to dY = f (X,Y)dX.- VII. Stratonovich and Itô integrals of first order forms.- 7.1 Symmetric differentiation of first order forms.- 7.3-7 Stratonovich integral of a first order form along a semimartingale.- 7.9-11 Interpolation rule.- 7.13 Existence of geodesic interpolation rule.- 7.14 Approximation of a Stratonovich integral by discretizing time.- 7.15 Stratonovich operator.- 7.16 Stratonovich stochastic differential equation ?Y = e(X,Y)?X.- 7.21 Existence and uniqueness of the solution to ?Y = e(X,Y)?X.- 7.24, 27 Approximating the solution to ?Y = e(X,Y)?X.- 7.28 Connections, interpreted in terms of second order geometry.- 7.31 Geodesies and martingales, characterized with purely second order vectors.- 7.33-34 Itô integral of a first order form.- 7.35 Characterization of martingales by Itô integrals.- 7.37 Discrete approximation of an Itô integral.- VIII. Parallel transport and moving frame.- 8.1 Parallel transport.- 8.5 Existence, uniqueness and linearity of parallel transport.- 8.6 Isometry of parallel transport.- 8.7 Geometric intepretation of connections.- 8.9 Stochastic parallel transport.- 8.13 Existence, uniqueness and linearity of stochastic parallel transport.- 8.14 Isometry of stochastic parallel transport.- 8.15 Discrete approximation of a stochastic parallel transport.- 8.17 Moving frame, parallel moving frame.- 8.18 Frame bundle.- 8.19 Itô depiction of a semimartingale in a moving frame.- 8.20 Stratonovich depiction of a semimartingale in a moving frame.- 8.21 Characterization of martingales by their Itô depiction.- 8.22-23 Lifting a semimartingale in the tangent space.- 8.24 A sufficient condition for ? ??,FdX? = ? ??,?X?.- 8.26 Characterization of geodesies, martingales and Brownian motions by their lifting.- 8.29-31 Development in M of a semimartingale in TxM.- Appendix: A short presentation of stochastic calculus.