2,002 research outputs found
Controlling Rough Paths
We formulate indefinite integration with respect to an irregular function as
an algebraic problem and provide a criterion for the existence and uniqueness
of a solution. This allows us to define a good notion of integral with respect
to irregular paths with Hoelder exponent greater than 1/3 (e.g. samples of
Brownian motion) and study the problem of the existence, uniqueness and
continuity of solution of differential equations driven by such paths. We
recover Young's theory of integration and the main results of Lyons' theory of
rough paths in Hoelder topology.Comment: 43 pages, no figures, corrected a proof in Sec.
A new definition of rough paths on manifolds
Smooth manifolds are not the suitable context for trying to generalize the
concept of rough paths on a manifold. Indeed, when one is working with smooth
maps instead of Lipschitz maps and trying to solve a rough differential
equation, one loses the quantitative estimates controlling the convergence of
the Picard sequence. Moreover, even with a definition of rough paths in smooth
manifolds, ordinary and rough differential equations can only be solved locally
in such case. In this paper, we first recall the foundations of the Lipschitz
geometry, introduced in "Rough Paths on Manifolds" (Cass, T., Litterer, C. &
Lyons, T.), along with the main findings that encompass the classical theory of
rough paths in Banach spaces. Then we give what we believe to be a minimal
framework for defining rough paths on a manifold that is both less rigid than
the classical one and emphasized on the local behaviour of rough paths. We end
by explaining how this same idea can be used to define any notion of coloured
paths on a manifold
Sensitivities via Rough Paths
Motivated by a problematic coming from mathematical finance, this paper is
devoted to existing and additional results of continuity and differentiability
of the It\^o map associated to rough differential equations. These regularity
results together with Malliavin calculus are applied to sensitivities analysis
for stochastic differential equations driven by multidimensional Gaussian
processes with continuous paths, especially fractional Brownian motions.
Precisely, in that framework, results on computation of greeks for It\^o's
stochastic differential equations are extended. An application in mathematical
finance, and simulations, are provided.Comment: 36 pages, 1 figur
Quasilinear SPDEs via rough paths
We are interested in (uniformly) parabolic PDEs with a nonlinear dependance
of the leading-order coefficients, driven by a rough right hand side. For
simplicity, we consider a space-time periodic setting with a single spatial
variable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - \sigma(u)f ) =0
\end{equation*} where is the projection on mean-zero functions, and
is a distribution and only controlled in the low regularity norm of for on the parabolic H\"older scale.
The example we have in mind is a random forcing and our assumptions
allow, for example, for an which is white in the time variable and
only mildly coloured in the space variable ; any spatial covariance
operator with is
admissible.
On the deterministic side we obtain a -estimate for , assuming
that we control products of the form and with solving
the constant-coefficient equation . As a
consequence, we obtain existence, uniqueness and stability with respect to of small space-time periodic solutions for small data. We
then demonstrate how the required products can be bounded in the case of a
random forcing using stochastic arguments.
For this we extend the treatment of the singular product via a
space-time version of Gubinelli's notion of controlled rough paths to the
product , which has the same degree of singularity but is
more nonlinear since the solution appears in both factors. The PDE
ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary
Schauder theory.Comment: 65 page
- …