Solving DSGE Models with a Nonlinear Moving Average

We introduce a nonlinear infinite moving average as an alternative to the standard state-space policy function for solving nonlinear DSGE models. Perturbation of the nonlinear moving average policy function provides a direct mapping from a history of innovations to endogenous variables, decomposes the contributions from individual orders of uncertainty and nonlinearity, and enables familiar impulse response analysis in nonlinear settings. When the linear approximation is saddle stable and free of unit roots, higher order terms are likewise saddle stable and first order corrections for uncertainty are zero. We derive the third order approximation explicitly and examine the accuracy of the method using Euler equation tests.Perturbation, nonlinear impulse response, DSGE, solution methods

Existence and Uniqueness of Perturbation Solutions to DSGE Models

We prove that standard regularity and saddle stability assumptions for linear approximations are sufficient to guarantee the existence of a unique solution for all undetermined coefficients of nonlinear perturbations of arbitrary order to discrete time DSGE models. We derive the perturbation using a matrix calculus that preserves linear algebraic structures to arbitrary orders of derivatives, enabling the direct application of theorems from matrix analysis to prove our main result. As a consequence, we provide insight into several invertibility assumptions from linear solution methods, prove that the local solution is independent of terms first order in the perturbation parameter, and relax the assumptions needed for the local existence theorem of perturbation solutions.Perturbation, matrix calculus, DSGE, solution methods, Bézout theorem; Sylvester equations

Integrated modeling and parallel computation of laser-induced axisymmetric rod growth

To fully investigate a pyrolytic Laser-induced chemical vapor deposition (LCVD) system for growing an axisymmetric rod, a novel integrated three-dimensional mathematical model was developed not only to describe the heat transport in the deposit and substrate, but also to simulate the gas-phase in the heated reaction zone and its effect on growth rate. The integrated model consists of three components: the substrate, rod, and gas-phase domains. Each component is a separate model and the three components are dynamically integrated into one model for simulating the iterative and complex process of rod deposition. The gas-phase reaction is modeled by the gas-phase component, an adaptive domain attached on the top part of the rod. Its size and mesh decomposition is dynamically determined by the rod temperature distribution and the chosen threshold. The temperature and molar ratio are predicted and used to adjust the growth rate, by taking into account the diffusion limited growth regime, and to improve the simulation of entire deposition process. The substrate component describes the heat flow into the substrate, and the substrate surface temperature can be used to predict the initial rod growth which may affect the successive growth of the rod. The rod growth process is simulated using a layer-by-layer axisymmetric model. For each layer, the rod grows along the outward normal direction at each point on the rod surface. This simplified model makes the process more predictable and easier to control by specifying the height of the rod and the number of total iterations. Finite difference schemes, iterative numerical methods, and parallel algorithms were developed for solving the model. The numerical computation is stable, convergent, and efficient. The model and numerical methods are implemented sequentially and in parallel using a standard C++ code and Message Passing Interface (MPI). The program can be easily installed and executed on different platforms, such as Unix and Windows XP. Computation in the gas-phase domain is encapsulated in a C++ class, and it is convenient for users to choose either the integrated or the kinetic model to perform simulation of rod growth. Parallel implementation improves the computational performance. To demonstrate the capability of the integrated model, silane is chosen as the precursor to grow the axisymmetric rod with silicon as deposit and graphite as substrate. The integrated 3D LCVD model and the corresponding numerical methods are applied to simulate the gas-phase reaction process, and to predict heat transfer, molar ratio, initial and successive rod growths and growth time at each iteration. It is found that the diffusion-limited growth can affect the deposition process and must be taken into account when the temperature is higher than a certain threshold. The initial rod growth can affect the successive rod growth and its geometry. This modeling approach may provide a useful means for investigating the effect of different model parameters for optimizing the LCVD process

New feature of low pTp_{T} charm quark hadronization in pppp collisions at s=7\sqrt{s}=7 TeV

Treating the light-flavor constituent quarks and antiquarks that can well describe the data of light-flavor hadrons in $pp$ collisions at $\sqrt{s}=7$ TeV as the underlying source of chromatically neutralizing the charm quarks of low transverse momenta ($p_{T}$), we show that the experimental data of $p_{T}$ spectra of single-charm hadrons $D^{0,+}$, $D^{*+}$ $D_{s}^{+}$, $\Lambda_{c}^{+}$ and $\Xi_{c}^{0}$ at mid-rapidity in the low $p_{T}$ range ($2\lesssim p_{T}\lesssim7$ GeV/$c$) in $pp$ collisions at $\sqrt{s}=7$ TeV can be well understood by the equal-velocity combination of perturbatively-created charm quarks and those light-flavor constituent quarks and antiquarks. This suggests a possible new scenario of low $p_{T}$ charm quark hadronization, in contrast to the traditional fragmentation mechanism, in $pp$ collisions at LHC energies. This is also another support for the exhibition of the effective constituent quark degrees of freedom for the small parton system created in $pp$ collisions at LHC energies.Comment: 7 pages, 5 figure