Nicolas Victoir

Differential equations driven by Gaussian signals

We consider multi-dimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of Lévy area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation results. Together with a new RKHS embedding, we obtain a powerful yet conceptually simple framework in which to analysize differential equations driven by Gaussian signals in the rough paths sense.

Asymmetric Cubature Formulae with Few Points in High Dimension for Symmetric Measures

Let

NON-DEGENERACY OF WIENER FUNCTIONALS ARISING FROM ROUGH DIFFERENTIAL EQUATIONS

mu

Levy area for the free Brownian motion: existence and non-existence

be a positive measure on

Multidimensional Stochastic Processes as Rough Paths: Theory and Applications

mathbb{R}^d

Analysis on local Dirichlet spaces

invariant under the group of reflections and permutations, and let m be a natural number. We describe a method to construct cubature formulae of degree m with respect to

Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs

mu

Approximations of the Brownian rough path with applications to stochastic analysis

, with n positive weights and n points in the support of

A note on the notion of geometric rough paths

mu

Euler Estimates for Rough Differential Equations

and such that n grows at most like dm with the dimension d. We apply this method to classical measures to explicitly construct cubature formulae of degree 5 with the number of points growing at most like d3.