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 Ito 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 Ito integral of a first order form.- 7.35 Characterization of martingales by Ito integrals.- 7.37 Discrete approximation of an Ito 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 Ito 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 Ito 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.
