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Martingales on manifolds and geometric Ito calculus
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Darling, R. W. R., 1954 (1982) Martingales on manifolds and geometric Ito calculus. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b1754582~S15
Abstract
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 coordinates. 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 nondegenerate diffusions on manifolds from the viewpoint of geometric Ito calculus, and of a Girsanovtype theorem due to Elworthy. Part B applies the methods of Part A to the study of 'Γmartingales' on M. It begins with six characterizations of Γmartingales, of which three are new; the simplest is: a process whose image under every local Γconvex function is (in a certain sense) a submartingale, However to obtain the other characterizations from this one requires a difficult proof. The behaviour of Γmartingales under harmonic maps, harmonic morphisms and affine maps is also studied. On a Riemannian manifold with a metric connection Γ, a Γmartingale is said to be L2 if its stochastic development is an L2 Γmartingale. We prove that if M is complete, then every such process has an almost sure limit, taking values in the onepoint 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 Weian.) The final chapter consists of a collection of examples of Γmartingales, e.g. on parallelizable manifolds such as Lie groups, and on surfaces embedded in R3. The final example is of a Γmartingale on the torus T (Γ is the LeviCivita connection for the embedded metric) which is also a martingale in R3.
Item Type:  Thesis or Dissertation (PhD) 

Subjects:  Q Science > QA Mathematics 
Library of Congress Subject Headings (LCSH):  Differentiable manifolds, Riemannian manifolds, Martingales (Mathematics), Semimartingales (Mathematics), Calculus 
Date:  January 1982 
Institution:  University of Warwick 
Theses Department:  Mathematics Institute 
Thesis Type:  PhD 
Publication Status:  Unpublished 
Supervisor(s)/Advisor:  Elworthy, K. D. 
Sponsors:  Science Research Council (Great Britain) (SRC) ; London Mathematical Society 
Extent:  xvi, 125 p. 
Language:  eng 
URI:  http://wrap.warwick.ac.uk/id/eprint/3948 
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