arXiv subjects

K. Vaninsky

Publications and source records attributed to K. Vaninsky.

4 recordsLinked to original sources

arXiv2012

Investor's sentiment in multi-agent model of the continuous double auction

We introduce and treat rigorously a new multi-agent model of the continuous double auction or in other words the order book (OB). It is designed to explain collective behaviour of the market when new information affecting the market arrives. The novel feature of the model is two additional slow changing parameters, the so-called sentiment functions. These sentiment functions measure the conception of the fair price of two groups of investors, namely, bulls and bears. Our model specifies differential equations for the time evolution of sentiment functions and constitutes a nonlinear Markov process which exhibits long term correlations. We explain the intuition behind equations for sentiment functions and present numerical simulations which show that the behaviour of our model is similar to the behaviour of the real market. We also obtain a diffusion limit of the model, the Ornstein-Uhlenbeck type process with variable volatility. The volatility is proportional to the difference of opinions of bulls and bears about the fair price of a security. The paper is complimentary to our previous work where mathematical proofs are presented.

q-fin.TR↗

arXiv2011

The AdaBoost Flow

We introduce a dynamical system which we call the AdaBoost flow. The flow is defined by a system of ODEs with control. We show that three algorithms of the AdaBoost family (i) the AdaBoost algorithm of Schapire and Freund (ii) the arc-gv algorithm of Breiman (iii) the confidence rated prediction of Schapire and Singer can be can be embedded in the AdaBoost flow. The nontrivial part of the AdaBoost flow equations coincides with the equations of dynamics of nonperiodic Toda system written in terms of spectral variables. We provide a novel invariant geometrical description of the AdaBoost algorithm as a gradient flow on a foliation defined by level sets of the potential function. We propose a new approach for constructing boosting algorithms as a continuous time gradient flow on measures defined by various metrics and potential functions. Finally we explain similarity of the AdaBoost algorithm with the Perelman's construction for the Ricci flow.

stat.ML↗

arXiv2002

The Atiyah--Hitchin bracket and the open Toda lattice

The dynamics of finite nonperiodic Toda lattice is an isospectral deformation of the finite three--diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one--to--one correspondence with their Weyl functions. These are rational functions mapping the upper half--plane into itself. We consider representations of the Weyl functions as a quotient of two polynomials and exponential representation. We establish a connection between these representations and recently developed algebraic--geometrical approach to the inverse problem for Jacobi matrix. The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We show that an invariance of the AH structure under linear--fractional transformations leads to two systems of canonical coordinates and two families of commuting Hamiltonians. We establish a relation of one of these systems with Jacobi elliptic coordinates.

math-ph↗

arXiv2001

The Atiyah-Hitchin Bracket for the Cubic Nonlinear Schrodinger Equation. I. General Potentials

This is the first in a series of papers on Poisson formalism for the cubic nonlinear Schr\"{o}dinger equation with repulsive nonlinearity and its relation to complex geometry. In this paper we study general continuous potentials. We demonstrate that the Weyl functions of the corresponding auxiliary Dirac spectral problem carry a natural Poisson structure. We call it the Atiyah--Hitchin Poisson bracket. We show that the Poisson bracket on the phase space is the image of the Atiyah--Hitchin bracket on Weyl functions under the inverse spectral transform.

math-ph↗