Advection and dispersion in time and space

Previous work showed how moving particles that rest along their trajectory lead to time-nonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions to the equation are obtained by subordination. The form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly.     

Fractional diffusion models of option prices in markets with jumps

Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Lévy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Lévy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derived.     

Fermionic-mode entanglement in non-Markovian environment

We evaluate the non-Markovian effects on the entanglement dynamics of a fermionic system interacting with two dissipative vacuum reservoirs. The exact solution of density matrix is derived by utilizing the Feynman–Vernon influence functional theory in the fermionic coherent state representation and the Grassmann calculus, which are valid for both the fermionic and bosonic baths, and their difference lies in the dependence of the parity of the initial states. The fermionic entanglement dynamics is presented by adding an additional restriction to the density matrix known as the superselection rules. Our analysis shows that the usual decoherence suppression schemes implemented in qubits systems can also be achieved for systems of identical fermions, and the initial state proves its importance in the evolution of fermionic entanglement. Our results provide a potential way to decoherence controlling of identical fermions.     

Modular Hypergeometric Residue Sums of Elliptic Selberg Integrals

It is shown that the residue expansion of an elliptic Selberg integral gives rise to an integral representation for a multiple modular hypergeometric series. A conjectural evaluation formula for the integral then implies a closed summation formula for the series, generalizing both the multiple basic hypergeometric ₈₇ sum of Milne-Gustafson type and the (one-dimensional) modular hypergeometric ₈₇ sum of Frenkel and Turaev. Independently, the modular invariance ensures the asymptotic correctness of our multiple modular hypergeometric summation formula for low orders in a modular parameter.     

Stock price dynamics and option valuations under volatility feedback effect

According to the volatility feedback effect, an unexpected increase in squared volatility leads to an immediate decline in the price–dividend ratio. In this paper, we consider the properties of stock price dynamics and option valuations under the volatility feedback effect by modeling the joint dynamics of stock price, dividends, and volatility in continuous time. Most importantly, our model predicts the negative effect of an increase in squared return volatility on the value of deep-in-the-money call options and, furthermore, attempts to explain the volatility puzzle. We theoretically demonstrate a mechanism by which the market price of diffusion return risk, or an equity risk-premium, affects option prices and empirically illustrate how to identify that mechanism using forward-looking information on option contracts. Our theoretical and empirical results support the relevance of the volatility feedback effect. Overall, the results indicate that the prevailing practice of ignoring the time-varying dividend yield in option pricing can lead to oversimplification of the stock market dynamics.     

Examples of Twisted Cyclic Cocycles from Covariant Differential Calculi

For two covariant differential *-calculi, the twisted cyclic cocycle associated with the volume form is represented in terms of commutators for some self-adjoint operator and some *-representation of the underlying *-algebra.     

Topics in data assimilation: Stochastic processes

Stochastic models with varying degrees of complexity are increasingly widespread in the oceanic and atmospheric sciences. One application is data assimilation, i.e., the combination of model output with observations to form the best picture of the system under study. For any given quantity to be estimated, the relative weights of the model and the data will be adjusted according to estimated model and data error statistics, so implementation of any data assimilation scheme will require some assumption about errors, which are considered to be random. For dynamical models, some assumption about the evolution of errors will be needed. Stochastic models are also applied in studies of predictability.

The formal theory of stochastic processes was well developed in the last half of the twentieth century. One consequence of this theory is that methods of simulation of deterministic processes cannot be applied to random processes without some modification. In some cases the rules of ordinary calculus must be modified.

The formal theory was developed in terms of mathematical formalism that may be unfamiliar to many oceanic and atmospheric scientists. The purpose of this article is to provide an informal introduction to the relevant theory, and to point out those situations in which that theory must be applied in order to model random processes correctly.     

Li_2O-Al_2O_3-SiO_2-P_2O_5系统微晶玻璃受控析晶分析

应用DTA,XRD,SEM等手段研究了Li2O_Al2O3_SiO2_P2O5系统微晶玻璃的析晶过程和相组成。采用模拟计算方法考察了不同的析晶条件。引入适量的Al2O3等难熔物组分来改变析晶条件及残留玻璃相的组成和粘度。通过抑制主晶相(Li2O.2SiO2)晶粒的生长,最终获得了晶粒尺寸为0.1~0.3μm的透明微晶玻璃。