Consistency of a method of moments estimator based on numerical solutions to asset pricing models

This paper considers the properties of estimators based on numerical solutions to a class of economic models. In particular, the numerical methods discussed are those applied in the solution of linear integral equations, specifically Fredholm equations of the second kind. These integral equations arise out of economic models in which endogenous variables appear linearly in the Euler equations, but for which easily characterized solutions do not exist. Tauchen and Hussey [24] have proposed the use of these methods in the solution of the consumption-based asset pricing model. In this paper, these methods are used to construct method of moments estimators where the population moments implied by a model are approximated by the population moments of numerical solutions. These estimators are shown to be consistent if the accuracy of the approximation is increased with the sample size. This result depends on the solution method having the property that the moments of the approximate solutions converge uniformly in the model parameters to the moments of the true solutions

A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind

AbstractIn this paper an iterative approach for obtaining approximate solutions for a class of nonlinear Fredholm integral equations of the second kind is proposed. The approach contains two steps: at the first one, we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation, solution of the discretized form converges to the exact solution of the problem. Following that, in the next step, solution of the discretized form is approximated by an iterative approach. We finally on some examples show the efficiency of the proposed approach

Optimal homotopy asymptotic and homotopy perturbation methods for linear mixed volterra-fredholm ıntegral equations

Bu çalışmada, karma Volterra-Fredholm integral denklemleri optimal homotopi asimptotik metod (OHAM) ve Homotopi Perturbasyon metodu (HPM) vasıtasıyla irdelenmiştir. Yaklaşımımız zamandan bağımsız ve basit hesaplamalar yolu ile tam çözüme oldukça yaklaşık çözümler veren bir yöntemdir. Bu iki yöntemin karşılaştırılması tartışılmıştır. OHAM yaklaşımının doğruluğu ve etkinliği HPM çözümleri ile dört örnek kullanılarak karşılaştırılmıştır. Sonuçlar OHAM ın HPM ye göre daha verimli ve esnek bir yöntem olduğunu göstermektedir.In this paper, we study the mixed Volterra-Fredholm integral equations of the second kind by means of optimal homotopy asymptotic method (OHAM) and Homotopy Perturbation method (HPM).Our approach is independent of time and contains simple computations with quite acceptable approximate solutions in which approximate solutions obtained by these methods are close to exact solutions. Comparison of these methods have been discussed. The accuracy and efficiency of OHAM approach with respect to Homotopy Perturbation method (HPM) is illustrated by presenting four test examples. The results indicate that the OHAM is very effective and flexible to use with respect to HPM

Piecewise linear dynamic systems with time delays

A new method of constructing periodic solutions for piecewise linear dynamic systems with time delays is investigated. Although the existence and the uniqueness of the periodic solution are guaranteed by well-known theorems, existing schemes for actually constructing the periodic solution are either purely formal or approximate. The idea of constructing the exact solution is first pursued with the linear delay systems. The formal representation of the solution to the linear problem is viewed as a system of Fredholm integral equations of the second kind. Since the matrix kernel for this system of integral equations is separable, the integral equation can be reduced to a system of algebraic equations involving certain integral moments of the initial function. These observations lead to a transfer relationship between two state vectors in the form of a matrix equation. Then the problem can be posed as either an initial value problem (if one is seeking the transient solution), or a periodic solution problem (if one is seeking the unknown initial data). This Fredholm Integral Equation Method is used effectively to construct periodic solutions to piecewise linear differential-difference equations. The periodic solutions are constructed from a cascaded product of matrix equations derived for each linear region. The stability of the periodic solution is determined by solving an associated eigenvalue problem. The periodic solution and its stability analysis are exact in the sense that the error induced by the truncation process in the Fredholm Integral Equation Method can be made exponentially small as the size of the transfer matrix is increased

First passage times of two-correlated processes: analytical results for the Wiener process and a numerical method for diffusion processes

Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show that these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times. The convergence of the method is theoretically proved for bivariate diffusion processes. We derive explicit expressions for these and other quantities of interest in the case of a bivariate Wiener process, correcting previous misprints appearing in the literature. Finally we illustrate the application of the method through a set of examples.Comment: 18 pages, 3 figure

Hägusad teist liiki integraalvõrrandid

Käesolevas doktoritöös on uuritud hägusaid teist liiki integraalvõrrandeid. Need võrrandid sisaldavad hägusaid funktsioone, s.t. funktsioone, mille väärtused on hägusad arvud. Me tõestasime tulemuse sileda tuumaga hägusate Volterra integraalvõrrandite lahendite sileduse kohta. Kui integraalvõrrandi tuum muudab märki, siis integraalvõrrandi lahend pole üldiselt sile. Nende võrrandite lahendamiseks me vaatlesime kollokatsioonimeetodit tükiti lineaarsete ja tükiti konstantsete funktsioonide ruumis. Kasutades lahendi sileduse tulemusi tõestasime meetodite koonduvuskiiruse. Me vaatlesime ka nõrgalt singulaarse tuumaga hägusaid Volterra integraalvõrrandeid. Uurisime lahendi olemasolu, ühesust, siledust ja hägusust. Ülesande ligikaudseks lahendamiseks kasutasime kollokatsioonimeetodit tükiti polünoomide ruumis. Tõestasime meetodite koonduvuskiiruse ning uurisime lähislahendi hägusust. Nii analüüs kui ka numbrilised eksperimendid näitavad, et gradueeritud võrke kasutades saame parema koonduvuskiiruse kui ühtlase võrgu korral. Teist liiki hägusate Fredholmi integraalvõrrandite lahendamiseks pakkusime uue lahendusmeetodi, mis põhineb kõigi võrrandis esinevate funktsioonide lähendamisel Tšebõšovi polünoomidega. Uurisime nii täpse kui ka ligikaudse lahendi olemasolu ja ühesust. Tõestasime meetodi koonduvuse ja lähislahendi hägususe.In this thesis we investigated fuzzy integral equations of the second kind. These equations contain fuzzy functions, i.e. functions whose values are fuzzy numbers. We proved a regularity result for solution of fuzzy Volterra integral equations with smooth kernels. If the kernel changes sign, then the solution is not smooth in general. We proposed collocation method with triangular and rectangular basis functions for solving these equations. Using the regularity result we estimated the order of convergence of these methods. We also investigated fuzzy Volterra integral equations with weakly singular kernels. The existence, regularity and the fuzziness of the exact solution is studied. Collocation methods on discontinuous piecewise polynomial spaces are proposed. A convergence analysis is given. The fuzziness of the approximate solution is investigated. Both the analysis and numerical methods show that graded mesh is better than uniform mesh for these problems. We proposed a new numerical method for solving fuzzy Fredholm integral equations of the second kind. This method is based on approximation of all functions involved by Chebyshev polynomials. We analyzed the existence and uniqueness of both exact and approximate fuzzy solutions. We proved the convergence and fuzziness of the approximate solution.https://www.ester.ee/record=b539569

Fredholm factorization of Wiener-Hopf scalar and matrix kernels

A general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the pape

Numerical solutions of a class of second order boundary value problems on using Bernoulli Polynomials

The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the approximate solutions of 2nd order boundary value problems. Here the Bernoulli polynomials over the interval [0, 1] are chosen as trial functions so that care has been taken to satisfy the corresponding homogeneous form of the Dirichlet boundary conditions in the Galerkin weighted residual method. In addition to that the given differential equation over arbitrary finite domain [a, b] and the boundary conditions are converted into its equivalent form over the interval [0, 1]. All the formulas are verified by considering numerical examples. The approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. A reliable good accuracy is obtained in all cases.Comment: 6 table