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Martingales on manifolds and geometric Ito calculus
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Darling, R. W. R. (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 (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Differentiable manifolds, Riemannian manifolds, Martingales (Mathematics), Semimartingales (Mathematics), Calculus  
Official Date:  January 1982  
Dates: 


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 
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