Related papers: A uniform estimate for rough paths
Path integrals represent a powerful route to quantization: they calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration.…
Rough sheets are two-parameter analogs of rough paths. In this work the theory of integration over functions of two parameters is extended to cover the case of irregular functions by developing an appropriate notion of rough sheet. The main…
We continue the approach in Part I \cite{duchong19} to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part II deals with driving…
We prove a rough It\^o formula for path-dependent functionals of $\alpha$-H\"older continuous paths for $\alpha\in(0,1)$. Our approach combines the sewing lemma and a Taylor approximation in terms of path-dependent derivatives.
Recently it was proved that the group of rough paths modulo tree-like equivalence is isomorphic to the corresponding signature group through the signature map S (a generalized notion of taking iterated path integrals). However, the proof of…
This paper establishes a comprehensive concentration theory for truncated signatures of Gaussian rough paths. The signature of a path, defined as the collection of all iterated integrals, provides a complete description of its geometric…
We investigate rough differential equations with a time-dependent reflecting lower barrier, where both the driving (rough) path and the barrier itself may have jumps. Assuming the driving signals allow for Young integration, we provide…
We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We…
We develop a unified second-order parameterized complexity theory for spaces of integrable functions. This generalizes the well-established case of second-order parameterized complexity theory for spaces of continuous functions.…
We study the Taylor expansion for the solutions of differential equations driven by $p$-rough paths with $p>2$. We prove a general theorem concerning the convergence of the Taylor expansion on a nonempty interval provided that the vector…
The cumulant representation of the Fourier path integral method is examined to determine the asymptotic convergence characteristics of the imaginary-time density matrix with respect to the number of path variables $N$ included. It is proved…
We develop a general framework for pathwise stochastic integration that extends F\"ollmer's classical approach beyond gradient-type integrands and standard left-point Riemann sums and provides pathwise counterparts of It\^o, Stratonovich,…
In this article we introduce the insertion method for reconstructing the path from its signature, i.e. inverting the signature of a path. For this purpose, we prove that a converging upper bound exists for the difference between the…
In this work we present the fundamental ideas of inference over paths, and show how this formalism implies the continuity equation, which is central for the derivation of the main partial differential equations that constitute…
The paper is split in two parts: in the first part, we construct the exact likelihood for a discretely observed rough differential equation, driven by a piecewise linear path. In the second part, we use this likelihood in order to construct…
We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform…
An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
In this article, I provide significant mathematical evidence in support of the existence of short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and…
A new formula for the probability that a standard Brownian motion stays between two linear boundaries is proved. A simple algorithm is deduced. Uniform precision estimates are computed. Different implementations have been made available…
We develop a fundamental framework for and extend the theory of rough paths to Lipschitz-gamma manifolds.