We present a novel criterion for the existence of Gaussian rough paths in the
sense of Friz{Victoir. It is formulated in terms of a covariance measure structure
together with a classical condition due to Jain{Monrad. It turns out that
this condition is easy to check in many examples which allows for a stochastic
calculus for a large class of Gaussian processes, ranging from (bi-)fractional
Brownian motion to processes given as random Fourier series.
We then discuss two applications. In our rst example we consider the concentration
of measure phenomenon on path spaces which can be described using
so{called transportation{cost inequalities. We show how rough path theory may
be used to establish such inequalities for the law of di usions. Our second example
deals with numerics for SDEs driven by Gaussian signals. Namely, we
present a multilevel Monte Carlo algorithm which can be applied when calculating
the mean of di usion functionals. We show how the multilevel approach
helps to reduce the computational complexity considerably compared to a standard
Monte Carlo approximation.
The rst part of the talk is based on joint work with Peter Friz (TU and
WIAS Berlin), Benjamin Gess (Universitat Bielefeld) and Archil Gulisashvili
(Ohio University). The last part is joint work with Christian Bayer (WIAS
Berlin), Peter Friz (TU and WIAS Berlin) and John Schoenmakers (WIAS
Berlin).
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