arXiv research

We provide a combinatorial condition characterizing curves that are short along a Teichmueller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to S x R are also short in the…

A short proof for curve lengths on hyperbolic surfaces.

In the context of multi-curve modeling we consider a two-curve setup, with one curve for discounting (OIS swap curve) and one for generating future cash flows (LIBOR for a give tenor). Within this context we present an approach for the clean-valuation pricing of FRAs and CAPs (linear and nonlinear derivatives) with one…

This paper corrects an error in [Keller-Ressel, M. and Steiner T. "Yield curve shapes and the asymptotic short rate distribution in affine one-factor models." Finance and Stochastics 12.2 (2008): 149-172]. The error concerns the correct expression for the boundary between normal and humped yield curve behavior in affin…

The recent financial crisis has led to so-called multi-curve models for the term structure. Here we study a multi-curve extension of short rate models where, in addition to the short rate itself, we introduce short rate spreads. In particular, we consider a Gaussian factor model where the short rate and the spreads are…

The study finds at least two short, simple geodesic chords on a disk with convex boundary.

By the curve shortening flow, the only closed embedded contracting self-similar solutions are circles: we give a very short and intuitive geometric proof of this basic and classical result using an idea of Gage.

For two measured laminations $ν^+$ and $ν^-$ that fill up a hyperbolizable surface $S$ and for $t \in (-\infty, \infty)$, let $L_t$ be the unique hyperbolic surface that minimizes the length function $e^t l(ν^+) + e^{-t} l(ν^-)$ on Teichmuller space. We characterize the curves that are short in $L_t$ and estimate their…

Study curve shortening flow on Riemann surfaces with conical singularities.

Scheme minimizes $p$-elastic energy of curves over time.

Kernel method estimates long-term effects from short-term data.

We prove short-time existence of φ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.

Geometric flow on curves in S^3 generates YO equations solutions.

Method constructs WP geodesics to study Teichmüller space behavior.

3D spheres can't be swept by short curves, complicating geodesic length estimates.

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle $α\in (0, π)$: The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class $W_2^γ$ with $γ\in (\tfrac{3}{2}, 2]$. For …

We consider a short rate model, driven by a stochastic process on the cone of positive semidefinite matrices. We derive sufficient conditions ensuring that the model replicates normal, inverse or humped yield curves.

We consider a model for interest rates, where the short rate is given by a time-homogenous, one-dimensional affine process in the sense of Duffie, Filipovic and Schachermayer. We show that in such a model yield curves can only be normal, inverse or humped (i.e. endowed with a single local maximum). Each case can be cha…

Sharp estimate shown to be rigid on curved surfaces.

In this short note we prove that in the case of elliptic curves, the isomorphism of generalized complex structure between $T$-dual manifolds described by Cavalcanti-Gualtieri coincides with the mirror map for elliptic curves described by Polishchuk and Zaslow.

Mirzakhani obtained the asymptotic growth, when $L\to\infty$, of the number of curves in the mapping class group orbit of some given simple curve and with length at most $L$. Years later she extended this result from simple to arbitrary curves. Here we give a short and relative low-tech argument showing how to derive t…

Study compares hyperbolic and extremal lengths for shortest curves.

We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated b…

Characterizes curves with short representatives on hyperbolic surfaces.

Study on ordering geodesics on hyperbolic surfaces.

We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curve…

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or HJM modeling, can be consolidated. We model a numeraire process and multiplicative spreads between Libor rates and simply compounded OIS rates as functions …

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in ge…

Nelson and Siegel curves are widely used to fit the observed term structure of interest rates in a particular date. By the other hand, several interest rate models have been developed such their initial forward rate curve can be adjusted to any observed data, as the Ho-Lee and the Hull and White one factor models. In t…

High-dimensional curved diffusions show abrupt convergence at a critical time.

Study improves prediction of commodity futures using multi-factor model.

In this short note we survey some results about the fundamental group of a compact negatively curved manifold. In particular, we review a theorem of Gusevskij, it states that the fundamental group of a compact negatively curved manifold does not belong to $\mathcal{C},$ where $\mathcal{C}$ is the smallest class of grou…

Study connects flow dynamics to 3D geometry via surface intersections.

For odd-dimensional spheres, there's always a second short geodesic.

Smoothly embed maps with controlled curvature errors.

Ancient curve shortening flows have entropy and curvature bounds equivalent.

We prove that any properly oriented $C^{2,1}$ isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable…

We study the evolution of closed inextensible planar curves under a second order flow that decreases the $p$-elastic energy. A short time existence result for $p \in (1,\infty)$ is obtained via a minimizing movements method. For $p = 2$, that is in the case of the classic elastic energy, long-time existence is retrieve…

We give a combinatorial proof, using the hyperbolicity of the curve graphs, of the bounded geodesic image theorem of Masur and Minsky. Recently it has been shown that curve graphs are uniformly hyperbolic, thus a universal bound can be given for the diameter of the geodesic image. We also generalize the theorem for pro…

We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve h…

The paper introduces a new short rate model with memory components.

We present an arbitrage-free non-parametric yield curve prediction model which takes the full (discretized) yield curve as state variable. We believe that absence of arbitrage is an important model feature in case of highly correlated data, as it is the case for interest rates. Furthermore, the model structure allows t…

Study curves evolving on hypersurfaces with free boundaries, preserving length.

Study of orbifold mapping class groups via arc and curve actions.

We prove that curves indicated in the title exist. This results answers to a question posed by A.G.Vitushkin about 30 years ago. We also discuss the minimal number of boundary components of a curve in the unit ball passing through the center, under the condition that all these components are shorter than a given number…

A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length $2π$. The estimate bounds the length of any chord from below in terms of the arc length between its endpoints and elapsed time. Applying the estimate to short segments we deduce directly …

We discuss a simple extension of the Ho and Lee model with generic time-dependent drift in which: 1) we compute bond prices analytically; 2) the yield curve is sensible and the asymptotic yield is positive; and 3) our analytical solution provides a clean and simple way of separating volatility from the drift in the sho…

Square-like quadrilaterals inscribed in space curves proven for finite total curvature.