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Vadim Linetsky - One of the best experts on this subject based on the ideXlab platform.

  • Positive Eigenfunctions of markovian pricing operators hansen scheinkman factorization ross recovery and long term pricing
    Operations Research, 2016
    Co-Authors: Likuan Qin, Vadim Linetsky
    Abstract:

    This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process, or BRP) and the stochastic discount factor (SDF) is a Positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a Positive Eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this Eigenfunction (recurrent Eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman factorization of the Markovian SDF corresponding to the recurrent Eigenfunction; extend the Recovery Theorem from discrete time, finite state irreducible Markov chains to recurrent BRPs; and obtain the long-maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient condition...

  • Positive Eigenfunctions of markovian pricing operators hansen scheinkman factorization ross recovery and long term pricing
    Research Papers in Economics, 2015
    Co-Authors: Likuan Qin, Vadim Linetsky
    Abstract:

    This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process (BRP)) and the stochastic discount factor (SDF) is a Positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a Positive Eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this Eigenfunction (recurrent Eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman (2009) factorization of the Markovian SDF corresponding to the recurrent Eigenfunction, extend the Recovery Theorem of Ross (2015) from discrete time, finite state irreducible Markov chains to recurrent BRPs, and obtain the long maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for existence of a recurrent Eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping.

Likuan Qin - One of the best experts on this subject based on the ideXlab platform.

  • Positive Eigenfunctions of markovian pricing operators hansen scheinkman factorization ross recovery and long term pricing
    Operations Research, 2016
    Co-Authors: Likuan Qin, Vadim Linetsky
    Abstract:

    This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process, or BRP) and the stochastic discount factor (SDF) is a Positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a Positive Eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this Eigenfunction (recurrent Eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman factorization of the Markovian SDF corresponding to the recurrent Eigenfunction; extend the Recovery Theorem from discrete time, finite state irreducible Markov chains to recurrent BRPs; and obtain the long-maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient condition...

  • Positive Eigenfunctions of markovian pricing operators hansen scheinkman factorization ross recovery and long term pricing
    Research Papers in Economics, 2015
    Co-Authors: Likuan Qin, Vadim Linetsky
    Abstract:

    This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process (BRP)) and the stochastic discount factor (SDF) is a Positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a Positive Eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this Eigenfunction (recurrent Eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman (2009) factorization of the Markovian SDF corresponding to the recurrent Eigenfunction, extend the Recovery Theorem of Ross (2015) from discrete time, finite state irreducible Markov chains to recurrent BRPs, and obtain the long maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for existence of a recurrent Eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping.

Solov’ev S. - One of the best experts on this subject based on the ideXlab platform.

  • Finite Element Approximation of the Minimal Eigenvalue and the Corresponding Positive Eigenfunction of a Nonlinear Sturm—Liouville Problem
    2020
    Co-Authors: Korosteleva D., Solov’ev P., Solov’ev S.
    Abstract:

    © 2019, Pleiades Publishing, Ltd. The problem of finding the minimal eigenvalue and the corresponding Positive Eigenfunction of the nonlinear Sturm—Liouville problem for the ordinary differential equation with coefficients nonlinear depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. A sufficient condition for the existence of a minimal eigenvalue and the corresponding Positive Eigenfunction of the nonlinear Sturm— Liouville problem is established. The original differential eigenvalue problem is approximated by the finite element method with Lagrangian finite elements of arbitrary order on a uniform grid. The error estimates of the approximate eigenvalue and the approximate Positive Eigenfunction to exact ones are proved. Investigations of this paper generalize well known results for the Sturm—Liouville problem with linear entrance on the spectral parameter

  • Finite Element Approximation of the Minimal Eigenvalue of a Nonlinear Eigenvalue Problem
    2020
    Co-Authors: Solov’ev S., Solov’ev P.
    Abstract:

    © 2018, Pleiades Publishing, Ltd. The problem of finding the minimal eigenvalue corresponding to a Positive Eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radiofrequency discharge at reduced pressures. A necessary and sufficient condition for the existence of a minimal eigenvalue corresponding to a Positive Eigenfunction of the nonlinear eigenvalue problem is established. The original differential eigenvalue problem is approximated by the finite element method on a uniform grid. The convergence of approximate eigenvalue and approximate Positive Eigenfunction to exact ones is proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter

  • Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem
    2020
    Co-Authors: Solov’ev S., Solov’ev P.
    Abstract:

    © 2014, Pleiades Publishing, Ltd. A condition for the existence of a minimum eigenvalue corresponding to a Positive Eigenfunction of the nonlinear eigenvalue problem for an ordinary differential equation is determined. The problem is approximated by a mesh scheme of the finite element method. The convergence of approximate solutions to exact ones is studied. Theoretical results are illustrated by numerical experiments for a model problem

Solov’ev P. - One of the best experts on this subject based on the ideXlab platform.

  • Finite Element Approximation of the Minimal Eigenvalue and the Corresponding Positive Eigenfunction of a Nonlinear Sturm—Liouville Problem
    2020
    Co-Authors: Korosteleva D., Solov’ev P., Solov’ev S.
    Abstract:

    © 2019, Pleiades Publishing, Ltd. The problem of finding the minimal eigenvalue and the corresponding Positive Eigenfunction of the nonlinear Sturm—Liouville problem for the ordinary differential equation with coefficients nonlinear depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. A sufficient condition for the existence of a minimal eigenvalue and the corresponding Positive Eigenfunction of the nonlinear Sturm— Liouville problem is established. The original differential eigenvalue problem is approximated by the finite element method with Lagrangian finite elements of arbitrary order on a uniform grid. The error estimates of the approximate eigenvalue and the approximate Positive Eigenfunction to exact ones are proved. Investigations of this paper generalize well known results for the Sturm—Liouville problem with linear entrance on the spectral parameter

  • Finite Element Approximation of the Minimal Eigenvalue of a Nonlinear Eigenvalue Problem
    2020
    Co-Authors: Solov’ev S., Solov’ev P.
    Abstract:

    © 2018, Pleiades Publishing, Ltd. The problem of finding the minimal eigenvalue corresponding to a Positive Eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radiofrequency discharge at reduced pressures. A necessary and sufficient condition for the existence of a minimal eigenvalue corresponding to a Positive Eigenfunction of the nonlinear eigenvalue problem is established. The original differential eigenvalue problem is approximated by the finite element method on a uniform grid. The convergence of approximate eigenvalue and approximate Positive Eigenfunction to exact ones is proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter

  • Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem
    2020
    Co-Authors: Solov’ev S., Solov’ev P.
    Abstract:

    © 2014, Pleiades Publishing, Ltd. A condition for the existence of a minimum eigenvalue corresponding to a Positive Eigenfunction of the nonlinear eigenvalue problem for an ordinary differential equation is determined. The problem is approximated by a mesh scheme of the finite element method. The convergence of approximate solutions to exact ones is studied. Theoretical results are illustrated by numerical experiments for a model problem

Nussbaum, Roger D. - One of the best experts on this subject based on the ideXlab platform.

  • Hidden Positivity and a New Approach to Numerical Computation of Hausdorff Dimension: Higher Order Methods
    2020
    Co-Authors: Falk, Richard S., Nussbaum, Roger D.
    Abstract:

    In [14], the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of Positive, linear, Perron-Frobenius operators $L_s$, an idea known in varying degrees of generality for many years. Although $L_s$ is not compact in the setting we consider, it possesses a strictly Positive $C^m$ Eigenfunction $v_s$ with eigenvalue $R(L_s)$ for arbitrary $m$ and all other points $z$ in the spectrum of $L_s$ satisfy $|z| \le b$ for some constant $b < R(L_s)$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $R(L_s) =1$. This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree $r$. Using an extension of the Perron theory of Positive matrices to matrices that map a cone $K$ to its interior and explicit a priori bounds on the derivatives of the strictly Positive Eigenfunction $v_s$, we give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge rapidly to $s_*$ as the mesh size decreases and/or the polynomial degree increases

  • $C^m$ Eigenfunctions of Perron-Frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension: Applications in $\mathbb{R}^1$
    2017
    Co-Authors: Falk, Richard S., Nussbaum, Roger D.
    Abstract:

    We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case that we consider here, our methods require only $C^3$ regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of Positive, linear, Perron-Frobenius operators $L_s$. The operators $L_s$ can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study $L_s$ in a Banach space of real-valued, $C^k$ functions, $k \ge 2$. We note that $L_s$ is not compact, but has essential spectral radius $\rho_s$ strictly less than the spectral radius $\lambda_s$ and possesses a strictly Positive $C^k$ Eigenfunction $v_s$ with eigenvalue $\lambda_s$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $\lambda_s =1$. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions. Using the theory of Positive linear operators and explicit a priori bounds on the derivatives of the strictly Positive Eigenfunction $v_s$, we give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge to $s_*$ as the mesh size approaches zero.Comment: This paper is a revised version of arXiv:1612.00870, which was the first part of a split of arXiv:1601.0673

  • A New Approach to Numerical Computation of Hausdorff Dimension of Iterated Function Systems: Applications to Complex Continued Fractions
    2017
    Co-Authors: Falk, Richard S., Nussbaum, Roger D.
    Abstract:

    In a previous paper, dealing with "Applications in $\mathbb{R}^1$," the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS and studied some applications in one dimension. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of Positive, linear, Perron-Frobenius operators $L_s$. In our context, $L_s$ is studied in a space of $C^m$ functions and is not compact. Nevertheless, it is has a strictly Positive $C^m$ Eigenfunction $v_s$ with Positive eigenvalue $\lambda_s$ equal to the spectral radius of $L_s$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $\lambda_s =1$. To compute the Hausdorff dimension of an IFS associated to complex continued fractions, (which may arise from an infinite iterated function system), we again approximate the eigenvalue problem by a collocation method, but now using continuous piecewise bilinear functions. Using the theory of Positive linear operators and explicit a priori bounds on the partial derivatives of the strictly Positive Eigenfunction $v_s$, we are able to give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge to $s_*$ as the mesh size approaches zero. We also demonstrate by numerical computations that improved estimates can be obtained by the use of higher order piecewise tensor product polynomial approximations, although the present theory does not guarantee that these are strict upper and lower bounds. An important feature of our approach is that it also applies to the much more general problem of computing approximations to the spectral radius of Positive transfer operators, which arise in many other applications.Comment: This paper is a revised version of arXiv:1612.00869, which is the second part of a split of arXiv:1601.0673