This article proposes a consistent and efficient estimator of the high-frequency covariance (quadratic covariation) of two arbitrary assets, observed asynchronously with market microstructure noise. This estimator is built on the marriage of the quasimaximum likelihood estimator of the quadratic variation and the proposed generalized synchronization scheme and thus is not influenced by the Epps effect. Moreover, the estimation procedure is free of tuning parameters or bandwidths and is readily implementable. Monte Carlo simulations show the advantage of this estimator by comparing it with a variety of estimators with specific synchronization methods. The empirical studies of six foreign exchange future contracts illustrate the time-varying correlations of the currencies during the 2008 global financial crisis, demonstrating the similarities and differences in their roles as key currencies in the global market.

No abstract uploaded!

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurston's geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure--a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Forman's expression of Ricci curvature on graphs.

For the viscous and heat-conductive fluids governed by the compressible NavierStokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the L^p - L^q estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L²-norm are obtained when the L¹-norm of the perturbation is bounded.

This paper is concerned with the Cauchy problem of the one-dimensional compressible Navier--Stokes equations with degenerate temperature dependent transport coefficients which satisfy conditions from the consideration in kinetic theory. A result on the existence and uniqueness of a globally smooth nonvacuum solution is obtained provided that the -norm of the initial perturbation) for some positive constant independent of . Here is the adiabatic gas constant. This is a Nishida--Smoller type global solvability result with large data.