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In this paper, we explore mutual information based stock networks to build regular vine copula structure on high frequency log returns of stocks and use it for the estimation of Value at Risk (VaR) of a portfolio of stocks. Our model is a data driven model that learns from a high frequency time series data of log returns of top 50 stocks listed on the National Stock Exchange (NSE) in India for the year 2014. The Ljung-Box test revealed the presence of Autocorrelation as well as Heteroscedasticity in the underlying time series data. Analysing the goodness of fit of a number of variants of the GARCH model on each working day of the year 2014, that is, 229 days in all, it was observed that ARMA(1,1)-EGARCH(1,1) demonstrated the best fit. The joint probability distribution of the portfolio is computed by constructed an R-Vine copula structure on the data with the mutual information guided minimum spanning tree as the key building block. The joint PDF is then fed into the Monte- Carlo simulation procedure to compute the VaR. If we replace the mutual information by the Kendall's Tau in the construction of the R-Vine copula structure, the resulting VaR estimations were found to be inferior suggesting the presence of non-linear relationships among stock returns. © 2021 Sharma, Sahni. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Mathematics

(SoNS) School of Natural Sciences

Fulltext

PLoS ONE

A mutual information based R-vine copula strategy to estimate VaR in high frequency stock market data

In this paper, we prove an extension of the Helson–Lowdenslager theorem for L2(T) for sub - Hilbert spaces under certain axioms. Furthermore, we provide an alternate proof of characterization of shift invariant sub-Hilbert spaces contained in H2(D) satisfying similar axioms. © 2021, Tusi Mathematical Research Group (TMRG).

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Advances in Operator Theory

On extension of Beurling–Helson–Lowdenslager theorem

This paper describes completely the invariant subspaces of the operator of multiplication by a finite Blaschke factor on the Banach space BMOA of analytic functions with bounded mean oscillation on the unit circle in the complex plane. As a simple application, we describe by very elementary means, the invariant subspaces of the co-analytic Toeplitz operator TB on H1. In the simplest case when B(z) = z, the invariant subspaces of TB on H1 were described by fairly deep arguments until the appearance of an elementary proof by two of the authors (Sahni &amp; Singh). In recent times, the common invariant subspaces of the operators of multiplication by B2 and B3, first in the case of z2 and z3, and then for an arbitrary finite Blaschke B, have proved to be critical in the context of Nevanlinna-Pick type interpolation on H2. Thus, keeping in mind the importance of invariant subspaces, we also offer a characterization of the common invariant subspaces of these operators on BMOA. Our proofs are that much more technical. Again, as an application, we obtain the common invariant subspaces of TB2 and TB3 on the Hardy space H1. © 2017, University at Albany. All rights reserved.

New York Journal of Mathematics

Invariance under finite blaschke factors on BMOA

In a recent paper, one of the authors - along with co-authors - extended the famous theorem of Beurling to the context of subspaces that are invariant under the class of subalgebras of H∞ of the form IH∞, where I is the inner function z2. In recent times, several researchers have replaced z2 by an arbitrary inner function I and this has proved important and fruitful in applications such as to interpolation problems of the Pick-Nevanlinna type. Keeping in mind the great deal of interest in such problems, in this paper, we provide analogues of the above mentioned IH∞ related extension of Beurling’s theorem in the setting of the Banach space BM OA, in the context of uniform algebras, on compact abelian groups with ordered duals and the Lebesgue space on the real line. We also provide a significant simplification of the proof of Beurling’s theorem in the setting of uniform algebras and a new proof of Helson’s generalization of Beurling’s theorem in the context of compact abelian groups with ordered duals. © 2016, University at Albany. All rights reserved.

Invariance under bounded analytic functions: Generalizing shifts

In this paper we characterize for 0 &lt; p ≤ ∞, the closed subspaces of Hp that are invariant under multiplication by all powers of a finite Blaschke factor B, except the first power. Our result clearly generalizes the invariant subspace theorem obtained by Paulsen and Singh [18] which has proved to be the starting point of important work on constrained Nevanlinna- Pick interpolation. Our method of proof can also be readily adapted to the case where the subspace is invariant under all positive powers of B (z) : The two results are in the mould of the classical Lax-Halmos Theorem and can be said to be Lax-Halmos type results in the finite multiplicity case for two commuting shifts and for a single shift respectively. © 2015 University of Houston.

Houston Journal of Mathematics

Lax-Halmos type theorems on Hp spaces

In the recent and important context of Nevanlinna-Pick interpolation theory for classes of holomorphic functions with constraints, see [4],[5], and [14], the starting point has been to describe the common invariant subspaces of the operators of multiplication by S2 and S3 on the Hardy classes. In the same spirit this paper offers as a new result the charcterization of the common invariant subspaces of S2 and S3 on the space BMOA. In addition we present a new, elementary and short proof of the invariant subspace characterization of the operator S on BMOA and use this to give a new proof of the invariant subspace characterization of the backward shift on the Hardy space H1 that is also short and elementary. © 2014 University of Houston.

Multiplication by monomials on BMOA

Regular and chaotic oscillations in a modified discrete two dimensional coupled predator-prey model, proposed recently, is re-investigated by observing the bifurcation diagram, calculating Lyapunov exponents and correlation dimensions for various orbits. The map evolves from one cycle to three cycles followed by period doubling and then to a chaotic regime and show bistability as its parameter λ varies, 0 < ≤ 1.211. Initial population size of the species and the value their coupling coefficients play crucial role for subsequent evolutionary phenomena of the system. Regular and chaotic evolutions depend completely on the coefficient ,λ, and on the initial prey and predator population. Lyapunov exponents and correlation dimensions provide true measure of chaos and also, to identify the chaotic orbits. Some recently discovered indicators, FLI, SALI and DLI have also been used to understand the nature of orbits.Mathematical analysis have been carried out to find stable and unstable fixed points and to obtain numerical value of the parameter λ for sensitive state of the system to initial conditions i.e. emergence of chaos. Bistability condition have been discussed for variation of λ values. Graphical representation of Lyapunov exponents and correlation dimensions provide better understanding of the nature of orbits which are further justified by the use of FLI, SALI and DLI.

Institute for Innovations & Inventions with Mathematics and IT

Applied Mathematical Sciences

Chaotic evaluations in a modified coupledLogistic type predator-prey model

Recently Yousefi and Hesameddini [13] have obtained a characterization for shift invariant subspaces of a special class of Hilbert spaces contained in the Hardy space H2. In the present note we settle an open problem posed by them in their paper. In fact, by discussing invariance under multiplication by finite Blaschke factors we prove a far more general result than the main result of [13]. We prove our results under much weaker assumptions than the assumptions of [13] and with a simpler proof. © 2011 The Japan Academy.