1,915 research outputs found
Multiscale Representations for Manifold-Valued Data
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere , the special orthogonal group , the positive definite matrices , and the Grassmann manifolds . The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the and maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as , , , where the and maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper
Functional central limit theorems for rough volatility
We extend Donsker's approximation of Brownian motion to fractional Brownian
motion with Hurst exponent and to Volterra-like processes. Some
of the most relevant consequences of our `rough Donsker (rDonsker) Theorem' are
convergence results for discrete approximations of a large class of rough
models. This justifies the validity of simple and easy-to-implement Monte-Carlo
methods, for which we provide detailed numerical recipes. We test these against
the current benchmark Hybrid scheme \cite{BLP15} and find remarkable agreement
(for a large range of values of~). This rDonsker Theorem further provides a
weak convergence proof for the Hybrid scheme itself, and allows to construct
binomial trees for rough volatility models, the first available scheme (in the
rough volatility context) for early exercise options such as American or
Bermudan.Comment: 30 pages, 11 figure
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