A Model of Market Limit Orders By Stochastic PDE's, Parameter Estimation, and Investment Optimization

In this paper we introduce a completely continuous and time-variate model of the evolution of market limit orders based on the existence, uniqueness, and regularity of the solutions to a type of stochastic partial differential equations obtained in Zheng and Sowers (2012). In contrary to several models proposed and researched in literature, this model provides complete continuity in both time and price inherited from the stochastic PDE, and thus is particularly suitable for the cases where transactions happen in an extremely fast pace, such as those delivered by high frequency traders (HFT's). We first elaborate the precise definition of the model with its associated parameters, and show its existence and uniqueness from the related mathematical results given a fixed set of parameters. Then we statistically derive parameter estimation schemes of the model using maximum likelihood and least mean-square-errors estimation methods under certain criteria such as AIC to accommodate to variant number of parameters . Finally as a typical economics and finance use case of the model we settle the investment optimization problem in both static and dynamic sense by analysing the stochastic (It\^{o}) evolution of the utility function of an investor or trader who takes the model and its parameters as exogenous. Two theorems are proved which provide criteria for determining the best (limit) price and time point to make the transaction

Magnetically Regulated Star Formation in Turbulent Clouds

We investigate numerically the combined effects of supersonic turbulence, strong magnetic fields and ambipolar diffusion on cloud evolution leading to star formation. We find that, in clouds that are initially magnetically subcritical, supersonic turbulence can speed up star formation, through enhanced ambipolar diffusion in shocks. The speedup overcomes a major objection to the standard scenario of low-mass star formation involving ambipolar diffusion, since the diffusion time scale at the average density of a molecular cloud is typically longer than the cloud life time. At the same time, the strong magnetic field can prevent the large-scale supersonic turbulence from converting most of the cloud mass into stars in one (short) turbulence crossing time, and thus alleviate the high efficiency problem associated with the turbulence-controlled picture for low-mass star formation. We propose that relatively rapid but inefficient star formation results from supersonic collisions of somewhat subcritical gas in strongly magnetized, turbulent clouds. The salient features of this shock-accelerated, ambipolar diffusion-regulated scenario are demonstrated with numerical experiments.Comment: 10 pages, 3 figures, accepted for publication in ApJ

Magnetic field twist driven by remote convective motions: Characteristics and twist rates

It is generally believed that convective motions below the solar photosphere induce a twist in the coronal magnetic field as a result of frozen-in physics. A question of interest is how much twist can one expect from a persistent convective motion, given the fact that dissipative effects will eventually figure. This question is examined by considering a model problem: two conducting plates, with finite resistivity, are set in sheared motion and forced at constant relative speed. A resistive plasma is between the plates and an initially vertical magnetic field connects the plates. The time rate of tilt experienced by the field is obtained as a function of Hartmann number and the resistivity ratio. Both analytical and numerical approaches are considered

Charge dynamics of the antiferromagnetically ordered Mott insulator

We study the charge dynamics of the half-filled Hubbard model on the square lattice at zero temperature. We employ a slave-fermion formulation in which the charge degrees of freedom are represented by fermionic holons and doublons and the bosonic spin degrees of freedom are assumed to be described by the antiferromagnetic Heisenberg model. The ground state has long-ranged magnetic (N\'eel) order and its Mott-insulating characteristics are the consequence of holon-doublon bound-state formation. Within this framework, we calculate the average double occupancy, the electron density of states, and the spectral function in the self-consistent Born approximation. The lower and upper Hubbard bands are reproduced, there is spectral-weight transfer into a coherent quasiparticle band at their lower edges, and the Mott gap, associated with holon-doublon binding, is renormalized due to the interactions of the holons and doublons with the magnons. We compute the zeros of the Green function at the chemical potential to discuss the Luttinger volume and show that the poles of the self-energy reflect the underlying quasiparticle dispersion with a spin-renormalized hopping parameter. Optical conductivity results relate the optical gap to the Mott gap of the insulating system. We conclude that a self-consistent treatment of the spin fluctuation effects on the charge degrees of freedom captures all the essential physics of the antiferromagnetic Mott-Hubbard insulator