Martingales on manifolds and geometric Ito calculus
This work studies properties of stochastic processes taking values in a differential manifold M with a linear connection Γ, or in a Riemannian manifold with a metric connection. Part A develops aspects of Ito calculus for semimartingales on M, using stochastic moving frames instead of local co-ordinates. New results include: -a formula for the Ito integral of a differential form along a semimartingale, in terms of stochastic moving frames and the stochastic development (with many useful corollaries); - an expression for such an integral as the limit in probability and in L2 of Riemann sums, constructed using the exponential map; - an intrinsic stochastic integral expression for the 'geodesic deviation', which measures the difference between the stochastic development and the inverse of the exponential map; -a new formulation of 'mean forward derivative' for a wide class of processes on M. Part A also includes an exposition of the construction of non-degenerate diffusions on manifolds from the viewpoint of geometric Ito calculus, and of a Girsanov-type theorem due to Elworthy. Part B applies the methods of Part A to the study of 'Γ-martingales' on M. It begins with six characterizations of Γ-martingales, of which three are new; the simplest is: a process whose image under every local Γ-convex function is (in a certain sense) a submartingale, However to obtain the other characterizations from this one requires a difficult proof. The behaviour of Γ-martingales under harmonic maps, harmonic morphisms and affine maps is also studied. On a Riemannian manifold with a metric connection Γ, a Γ-martingale is said to be L2 if its stochastic development is an L2 Γ-martingale. We prove that if M is complete, then every such process has an almost sure limit, taking values in the one-point compactification of M. No curvature conditions are required. (After this result was announced, a simpler proof was obtained by P. A. Meyer, and a partial converse by Zheng Wei-an.) The final chapter consists of a collection of examples of Γ-martingales, e.g. on parallelizable manifolds such as Lie groups, and on surfaces embedded in R3. The final example is of a Γ-martingale on the torus T (Γ is the Levi-Civita connection for the embedded metric) which is also a martingale in R3.