Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConvergence of trapezoid rule to rough integrals
72
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough integrals are usually Riemann-Stieltjes integrals with correction terms that are sometimes seen as unnatural. As opposed to those somewhat artificial correction terms, our endeavor in this note is to produce a trapezoid rule for rough integrals driven by general $d$-dimensional Gaussian processes. Namely we shall approximate a generic rough integral $int y , dX$ by Riemann sums avoiding the usual higher order correction terms, making the expression easier to work with and more natural. Our approximations apply to all controlled processes $y$ and to a wide range of Gaussian processes $X$ including fractional Brownian motion with a Hurst parameter $H>1/4$. As a corollary of the trapezoid rule, we also consider the convergence of a midpoint rule for integrals of the form $int f(X) dX$.
rate research
Read More
We prove that SU(N) bosonic Yang-Mills matrix integrals are convergent for dimension (number of matrices) $Dge D_c$. It is already known that $D_c=5$ for N=2; we prove that $D_c=4$ for N=3 and that $D_c=3$ for $Nge 4$. These results are consistent with the numerical evaluations of the integrals by Krauth and Staudacher.
We study the focusing mass-critical rough nonlinear Schroedinger equations, where the stochastic integration is taken in the sense of controlled rough path. We obtain the global well-posedness if the mass of initial data is below that of the ground state. Moreover, the existence of minimal mass blow-up solutions is also obtained in both dimensions one and two. In particular, these yield that the mass of ground state is exactly the threshold of global well-posedness and blow-up of solutions in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schroedinger equations with lower order perturbations.
A well-known theorem of Rodin & Sullivan, previously conjectured by Thurston, states that the circle packing of the intersection of a lattice with a simply connected planar domain $Omega$ into the unit disc $mathbb{D}$ converges to a Riemann map from $Omega$ to $mathbb{D}$ when the mesh size converges to 0. We prove the analogous statement when circle packings are replaced by the square tilings of Brooks et al.
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 $alpha >1/p$.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
