Preface -- Contents -- Chapter 1 Formal Stochastic Differential Equations -- 1.1 Motivation -- 1.2 The signature of a Brownian motion -- 1.3 The Chen-Strichartz development formula -- 1.4 Expectation of the signature of a Brownian motion -- 1.5 Expectation of the signature of other processes -- Chapter 2 Stochastic Differential Equations and Carnot Groups -- 2.1 The commutative case -- 2.2 Two-step nilpotent SDE's -- 2.3 N-step nilpotent SDE's -- 2.4 Pathwise approximation of solutions of SDEs -- 2.5 An introduction to rough paths theory -- Chapter 3 Hypoelliptic Flows -- 3.1 Hypoelliptic operators and Hörmander's theorem -- 3.2 Sub-Riemannian geometry -- 3.3 The tangent space to a hypoelliptic diffusion -- 3.4 Horizontal diffusions -- 3.5 Regular sublaplacians on compact manifolds -- 3.6 Stochastic differential equations driven by loops -- Appendix A Basic Stochastic Calculus -- A.1 Stochastic processes and Brownian motion -- A.2 Markov processes -- A.3 Martingales -- A.4 Stochastic integration -- A.5 Itô's formula -- A.6 Girsanov's theorem -- A.7 Stochastic differential equations -- A.8 Diffusions and partial differential equations -- A.9 Stochastic flows -- A.10 Malliavin calculus -- A.11 Stochastic calculus on manifolds -- Appendix B Vector Fields, Lie Groups and Lie Algebras -- B.1 Vector fields and exponential mapping -- B.2 Lie derivative of tensor fields along vector fields -- B.3 Exterior forms and exterior derivative -- B.4 Lie groups and Lie algebras -- B.5 The Baker-Campbell-Hausdorff formula -- B.6 Nilpotent Lie groups -- B.7 Free Lie algebras and Hall basis -- B.8 Basic Riemannian geometry -- Bibliography -- Index.