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Semimartingales and rough path theory; with A. Lejay.(
dec 2004)
We prove that the theory of rough paths,
which is used to define path-wise integrals and path-wise differential
equations,
can be used with continuous semi-martingales. We provide then an almost
sure theorem of
type Wong-Zakai. Moreover, we show that the conditions UT and
UCV, used to prove that one can interchange limits and Ito or Stratonovich
integrals, provide
the same result when one uses the rough paths theory.
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Volterra Bridges and Applications ; with F. Baudoin(
dec 2004)
We introduce and study the bridges over a given Volterra process.
As an application, we show that if a Gaussian process admits in
law a Volterra type representation, then this representation is
never unique.
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Good Rough Path Sequences ans Applications to Anticipativing
& Fractional Stochastic Calculus; with Peter Friz, Nicolas
Victoir ( jv 2005)
We consider anticipative Stratonovich stochastic differential equations
driven by some stochastic process (not necessarily a semi-martingale).
No
adaptedness of initial point or vector fields is assumed. Under a simple
condition on the stochastic process, we show that the unique solution
of the
above SDE understood in the rough path sense is actually a Stratonovich
solution. This condition is satisfied by the Brownian motion and the
fractional Brownian motion with Hurst parameter greater than $1/4$.
As
application, we obtain rather flexible results such as support theorems,
large deviation principles and Wong-Zakai approximations for SDEs driven
by
fractional Brownian Motion along anticipating vectorfields. In particular,
this unifies many results on anticipative SDEsW