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Register a new userDiscrete rough paths and limit theorems
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In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to limit theorems for weighted sums via rough path techniques. As a by-product, we provide a natural explanation of the various new asymptotic behaviors in contrast with the classical unweighted random sum case. We apply our principle to derive some weighted type Breuer-Major theorems, which generalize previous results to random sums that do not have to be in a finite sum of chaos. In this context, a Breuer-Major type criterion in notion of Hermite rank is obtained. We also consider some applications to realized power variations and to Itos formulas in law. In the end, we study the asymptotic behavior of weighted quadratic variations for some multi-dimensional Gaussian processes.
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Calculus via regularizations and rough paths are two methods to approach stochastic integration and calculus close to pathwise calculus. The origin of rough paths theory is purely deterministic, calculus via regularization is based on deterministic techniques but there is still a probability in the background. The goal of this paper is to establish a connection between stochastically controlled-type processes, a concept reminiscent from rough paths theory, and the so-called weak Dirichlet processes. As a by-product, we present the connection between rough and Stratonovich integrals for c{`a}dl{`a}g weak Dirichlet processes integrands and continuous semimartingales integrators.
We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that the solution map associated to differential equations driven by rough paths is a locally Lipschitz continuous map on the Sobolev rough path space for any arbitrary low regularity $alpha$ and integrability $p$ provided $alpha >1/p$.
We prove that all finite joint distributions of creation and annihilation operators in Monotone and anti-Monotone Fock spaces can be realized as Quantum Central Limit of certain operators on a $C^*$-algebra, at least when the test functions are Riemann integrable. Namely, the approximation is given by weighted sequences of creators and annihilators in discrete monotone $C^*$-algebras, the weight being the above cited test functions. The construction is then generalized to processes by an invariance principle.
We develop the rough path counterpart of It^o stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (It^o / forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.
We explore an asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.
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