Martingales on manifolds and geometric Ito calculus
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
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.
|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|
|Official Date:||January 1982|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Supervisor(s)/Advisor:||Elworthy, K. D.|
|Sponsors:||Science Research Council (Great Britain) (SRC) ; London Mathematical Society|
|Extent:||xvi, 125 p.|
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