Oscillation of Non-Linear Systems Close to Equilibrium Position in the Presence of Coarse-Graining in Time and Space

One considers the motion of nonlinear systems close to their equilibrium positions in the presence of coarse-graining in time on the one hand, and coarse-graining in time on the other hand. By considering a coarse-grained time as a time in which the increment is not dt but rather (dt)c > dt, one is led to introduce a modeling in terms of fractional derivative with respect to time; and likewise for coarse-graining with respect to the space variable x. After a few prerequisites on fractional calculus via modified Riemann-Liouville derivative, one examines in a detailed way the solutions of fractional linear differential equations in this framework, and then one uses this result in the linearization of nonlinear systems close to their equilibrium positions

Mandelbrot's stochastic time series models

I survey and illustrate the main time series models that Mandelbrot introduced into time series analysis in the 1960s and 1970s. I focus particularly on the members of the additive fractional stable family including Lévy flights and fractional Brownian motion (fBm), noting some of the less well‐known aspects of this family, such as the cases when the self‐similarity exponent H and the Hurst exponent J differ. I briefly discuss the role of multiplicative models in modeling the physics of cascades. I then recount the still little‐known story of Mandelbrot's work on fractional renewal models in the late 1960s, explaining how these differ from their more familiar fBm counterpart and form a “missing link” between fBm and the problem of random change points. I conclude by highlighting the frontier problem of damped fractional models

Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space (RKHS) $\mathscr{H}_{F}$. Why extensions? In science, experimentalists frequently gather spectral data in cases when the observed data is limited, for example limited by the precision of instruments; or on account of a variety of other limiting external factors. Given this fact of life, it is both an art and a science to still produce solid conclusions from restricted or limited data. In a general sense, our monograph deals with the mathematics of extending some such given partial data-sets obtained from experiments. More specifically, we are concerned with the problems of extending available partial information, obtained, for example, from sampling. In our case, the limited information is a restriction, and the extension in turn is the full positive definite function (in a dual variable); so an extension if available will be an everywhere defined generating function for the exact probability distribution which reflects the data; if it were fully available. Such extensions of local information (in the form of positive definite functions) will in turn furnish us with spectral information. In this form, the problem becomes an operator extension problem, referring to operators in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we have stressed hands-on-examples. Extensions are almost never unique, and so we deal with both the question of existence, and if there are extensions, how they relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text overlap with arXiv:1401.478

A full asymptotic series of European call option prices in the SABR model with beta=1

We develop two pricing formulae for European options in the SABR model with beta= 1 case by means of Malliavin Calculus. We follow the approach of Alòs et al (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call Dyson series on the return\u27s idiosyncratic noise yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor\u27s (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion.The second approach, which we call Dyson series on the common noise necessitates the calculation of only a one-dimensional integral, but the formula is more complex

An analysis of spending behaviour under liquidity constraints with an application to financial hedging

Imperial Users onl

Bayesian Pose Graph Optimization via Bingham Distributions and Tempered Geodesic MCMC

We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm for initializing pose graph optimization problems, arising in various scenarios such as SFM (structure from motion) or SLAM (simultaneous localization and mapping). TG-MCMC is first of its kind as it unites asymptotically global non-convex optimization on the spherical manifold of quaternions with posterior sampling, in order to provide both reliable initial poses and uncertainty estimates that are informative about the quality of individual solutions. We devise rigorous theoretical convergence guarantees for our method and extensively evaluate it on synthetic and real benchmark datasets. Besides its elegance in formulation and theory, we show that our method is robust to missing data, noise and the estimated uncertainties capture intuitive properties of the data.Comment: Published at NeurIPS 2018, 25 pages with supplement

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

STOCHASTIC EQUATIONS WITH FRACTIONAL NOISE: CONTINUITY IN LAW AND APPLICATIONS

The objective of the thesis is to study some properties and applications of stochastic equations driven by a fractional Brownian motion with Hurst parameter H. I n particular, we study the continuity with respect to H of the heat and wave multiplicative and additive stochastic partial differential equations driven by a noise which is white in the time variable and behaves like a fractional Brownian motion in the space variable. Morevoer, we study an analogous problem for a class of one-dimensional stochastic differential equations driven by a fractional noise, in the setting of rough paths theory. On the side of applications, we define and evaluate a stochastic model with the objective of forecasting the future electricity prices in the italian market. This model includes as the main stochastic component an equation driven by a fractional Brownian motion, plus a jump component which shows self-exciting properties, namely a Hawkes process