Eigenvalues Of Self-adjoint Operators Research Articles

Modelling of derivatives pricing using methods of spectral analysis

In this article expands the method of finding the approximate price for a wide class of derivative financial instruments. Using the spectral theory of self-adjoint operators in Hilbert space and the wave theory of singular and regular perturbations, the analytical formula of the approximate asset price is established. Methods for calculating the approximate price of options using the tools of spectral analysis, singular and regular wave theory in the case of fast and slow factors are developed. Combining methods from the spectral theory of singular and regular perturbations, it is possible to estimate the price of derivative financial instruments as a schedule by eigenfunctions. The approximate value of securities and their rate of return are calculated. Applying the theory of Sturm-Liouville, Fredholm’s alternative and analysis of singular and regular perturbations at different time scales have enabled us to obtain explicit formulas for the approximate value of securities and their yield on the basis of the development of their eigenfunctions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations. The theorem of closeness estimates for bond prices approximation is proved. An algorithm for calculating the approximate price of derivatives and the accuracy of estimates has been developed, which allows to analyze and draw precautionary conclusions and suggestions to minimize the risks of pricing derivatives that arise in the stock market. A model for finding the value of derivatives corresponding to the dynamics of the stock market and the size of financial flows has been developed. This model allows you to find the prices of derivatives and their volatility, as well as minimize speculative changes in pricing, analyze the progress of stock market processes and take concrete steps to improve the situation to optimize financial strategies. The used methodology of European options pricing based on the study of volatility behavior and analysis of the yield of financial instruments allows to increase the accuracy of the forecast and make sound management strategic decisions by stock market participants.

Finding the derivative price using the Vasicek model with multidimensional stochastic volatility

Methods of calculating the approximate price of options using instruments of spectral analysis, singular and regular wave theory in the context of influence of fast and slow acting factors are developed. By combining methods from the spectral theory of singular and regular disturbances, one can approximate the price of derivative financial instruments as a schedule of its own functions. The article uses the theory of spectral analysis and the singular and regular theory of perturbations, which are applied to the short-term interest rates described by the Vasicek model with multidimensional stochastic volatility. The approximate price of derivatives and their profitability are calculated. Applying the Sturm-Liouville theory, the Fredholm alternative, and the analysis of singular and regular disturbances in different time scales, explicit formulas were obtained for the approximation of bond prices and yields based on the development of their own functions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations. The theorem for estimating the accuracy of derivatives price approximation is established. Such a technique, in contrast to existing ones, makes it possible to study the stock market dynamics and to monitor the financial flows in the market. This greatly facilitates the statistical evaluation of their parameters in the process of monitoring the derivatives pricing and the study of volatility behavior for the profitability analysis and taking strategic management decisions on the stock market transactions.

CEV MODEL WITH STOCHASTIC VOLATILITY

This paper develops a systematic method for calculating approximate prices for a wide range of securities implying the tools of spectral analysis, singular and regular perturbation theory. Price options depend on stochastic volatility, which may be multiscale, in the sense that it may be driven by one fast-varying and one slow-varying factor. The found the approximate price of two-barrier options with multifactor volatility as a schedule for own functions. The theorem of estimation of accuracy of approximation of option prices is established. Explicit formulas have been found for finding the value of derivatives based on the development of eigenfunctions and eigenvalues of self-adjoint operators using boundary-value problems for singular and regular perturbations. This article develops a general method of obtaining a guide price for a broad class of securities. A general theory of derivative valuation of options generated by diffusion processes is developed. The algorithm of calculating the approximate price is given. The accuracy of the estimates is established. The theory developed is applied to a diffusion operator, which is decomposed by eigenfunctions and eigenvalues. The purpose of the article is to develop an algorithm for finding the approximate price of two-barrier options and to find explicit formulas for finding the value of derivatives based on the development of self-functions and eigenvalues of self-adjoint operators using boundary-value problems for singular and regular perturbations. Price finding is reduced to the problem solving of eigenvalues and eigenfunctions of a certain equation. The main advantage of our pricing methodology is that, by combining methods in spectral theory, regular perturbation theory, and singular perturbation theory, we reduce everything to equations to find eigenfunctions and eigenvalues.

Application of the spectral theory and perturbation theory to the study of Ornstein-Uhlenbeck processes

The theoretical bases of this paper are the theory of spectral analysis and the theory of singular and regular perturbations. We obtain an approximate price of Ornstein-Uhlenbeck double barrier options with multidimensional stochastic diffusion as expansion in eigenfunctions using infinitesimal generators of a $(l+r+1)$-dimensional diffusion in Hilbert spaces. The theorem of accuracy estimation of options prices approximation is established. We also obtain explicit formulas for derivatives price based on the expansion in eigenfunctions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations.

Local two-sided bounds for eigenvalues of self-adjoint operators

We examine the equivalence between an extension of the Lehmann–Maehly–Goerisch method developed a few years ago by Zimmermann and Mertins, and a geometrically motivated method developed more recently by Davies and Plum. We establish a general framework which allows sharpening various previously known results in these two settings and determine explicit convergence estimates for both methods. We demonstrate the applicability of the method of Zimmermann and Mertins by means of numerical tests on the resonant cavity problem.

Computability of the Spectrum of Self-Adjoint Operators and the Computable Operational Calculus

Self-adjoint operators and their spectra play a crucial rôle in analysis and physics. Therefore, it is a natural question whether the spectrum of a self-adjoint operator and its eigenvalues can be computed from a description of the operator. We prove that given a “program” of the operator one can obtain positive information on the spectrum as a compact set in the sense that a dense subset of the spectrum can be enumerated and a bound on the set can be computed. This generalizes some non-uniform results obtained by Pour-El and Richards, which imply that the spectrum of any computable self-adjoint operator is a recursively enumerable compact set. Additionally, we show that the eigenvalues of self-adjoint operators can be computed in the sense that we can compute a list of indices such that those elements of the already computed dense subset of the spectrum, whose indices are not enumerated in this list, form the set of eigenvalues. Beside these main results we prove some computability results about the operational calculus, which we need in our proofs.

Methods for computing lower bounds to eigenvalues of self-adjoint operators

New approaches for computing tight lower bounds to the eigenvalues of a class of semibounded self-adjoint operators are presented that require comparatively little a priori spectral information and permit the effective use of (among others) finite-element trial functions. A variant of the method of intermediate problems making use of operator decompositions having the form\(T^*T\) is reviewed and then developed into a new framework based on recent inertia results in the Weinstein-Aronszajn theory. This framework provides greater flexibility in analysis and permits the formulation of a final computational task involving sparse, well-structured matrices. Although our derivation is based on an intermediate problem formulation, our results may be specialized to obtain either the Temple-Lehmann method or Weinberger's matrix method.

Radically simplified proofs of the Weyl and Stenger inequalities concerning eigenvalues of self-adjoint operators

Radically simplified proofs of the generalized Weyl inequality and the Stenger inequality for eigenvalues of self-adjoint operators is presented. The main application of these results lie in atomic and molecular physics. The generalized inequality has application in obtaining bounds on molecular energy levels in cases where the molecular hamiltonian can be written as a sum of several atomic ones and the Stenger inequality has application in putting bounds on energy levels of atomic and molecular resonances to which the Feshbach formalism applies.

Continuity of the eigenvalues of self-adjoint operators with respect to the strong operator topology

Let Tn, T, and S be self-adjoint operators such that Tn converges to T in the sense of strong resolvent convergence. If S is bounded from below, σe(S) ∩ (−∞,O) = O, and Tn ≧ S for all n, then the negative eigenvalues of Tn converge to the negative eigenvalues of T. The corresponding eigenfunctions converge in norm. This generalizes a result due to T. Kato, where Tn ≧ T is assumed, and a recent result of the author for the case Tn ≦ Tn+1.

A variational principle for eigenvalue bounds

A functional is given which can be used to derive lower and upper bounds for the discrete eigenvalues of self-adjoint operator. The lower bounds of the functional are deduced and the connections to other variational principles are pointed out.

A coordinate-free treatment of the minimax and maximini theorems for eigenvalues of self-adjoint operators on a Hilbert space

The power and elegance of coordinate-free analysis is demonstrated by establishing the minimax and maximini theorems for eigenvalues of self-adjoint operators on a Hilbert space as trivial corollaries of a series of lemmas of astounding simplicity. Furthermore the analysis brings out the essential identity of the two results in contradistinction to what has been generally believed before. A host of other related results are also established and in particular the authors propose new simplified proofs of the inequalities of Weyl, Poincare and Bateman; the latter two of these are better known to physicists as the Hylleraas-Undheim and MacDonald theorems and are extensively used in establishing upper bound and convergence properties of energy eigenvalues of stationary states of quantum systems.

Error estimates for the computation of eigenvalues of self-adjoint operators

A convergence theorem is given for the numerical computation of eigenvalues of self-adjoint operators. The computed eigenvectors are used to construct trial solutions which can be used to obtain easily computable and accurate error estimates.