# Calcul Stochastique Covariant à Sauts & Calcul Stochastique à Sauts Covariants

Abstract : We propose a stochastic covariant calculus for
càdlàg semimartingales in the tangent bundle $TM$ over a manifold $M$.
A connection on $M$ allows us to define an intrinsic derivative of
a $C^1$ curve $(Y_t)$ in $TM$, the covariant
derivative. More precisely, it is the derivative of
$(Y_t)$ seen in a frame moving parallelly along its projection curve
$(x_t)$ on $M$. With the transfer principle, Norris defined the
stochastic covariant integration along a continuous semimartingale in
$TM$. We describe the case where the semimartingale jumps in $TM$,
using Norris's work and Cohen's results about stochastic calculus
with jumps on manifolds. We see that, depending on the order in
which we compose the function giving the jumps and the connection, we
obtain a (\it stochastic covariant calculus with jumps) or a (\it
stochastic calculus with covariant jumps).
Both depend on the choice of the connection and of the tools
(interpolation and connection rules) describing the
jumps in the meaning of Stratonovich or Itô. We study the choices
that make equivalent the two calculus. Under suitable conditions,
we recover Norris's results when $(Y_t)$ is continuous.
The continuous case is described by a covariant continuous calculus of
order two, a formalism defined with the notion of connection of order two.
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Document type :
Theses
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Cited literature [12 references]

https://tel.archives-ouvertes.fr/tel-00004226
Contributor : Laurence Maillard-Teyssier <>
Submitted on : Tuesday, January 20, 2004 - 12:57:55 PM
Last modification on : Friday, February 22, 2019 - 1:15:35 AM
Long-term archiving on : Friday, April 2, 2010 - 7:03:45 PM

### Identifiers

• HAL Id : tel-00004226, version 1

### Citation

Laurence Maillard-Teyssier. Calcul Stochastique Covariant à Sauts & Calcul Stochastique à Sauts Covariants. Mathematics [math]. Université de Versailles-Saint Quentin en Yvelines, 2003. English. ⟨tel-00004226⟩

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