Properties of solutions of stochastic differential equations driven by the G-Brownian motion

In this paper, we study the differentiability of solutions of stochastic differential equations driven by the $G$-Brownian motion with respect to the initial data and the parameter. In addition, the stability of solutions of stochastic differential equations driven by the $G$-Brownian motion is obtained

Testing and finding the generating functions of an option pricing mechanism through market data

We study dynamic pricing mechanisms of financial derivatives. A typical model of such pricing mechanism is the so-called g-expectation defined by solutions of a backward stochastic differential equation with g as its generating function. Black-Scholes pricing model is a special linear case of this pricing mechanism. We are mainly concerned with two types of pricing mechanisms in an option market: the market pricing mechanism through which the market prices of options are produced, and the ask-bid pricing mechanism operated through the system of market makers. The later one is a typical nonlinear pricing mechanism. Data of prices produced by these two pricing mechanisms are usually quoted in an option market. We introduce a criteria to test if a dynamic pricing mechanism under investigation is a g-pricing mechanism. This domination condition was statistically tested using CME data documents. The result of test is significantly positive. We also provide some useful characterizations of a pricing mechanism by its generating function

Entropy Injection as a Global Feedback Mechanism

Both preheating of the intergalactic medium and radiative cooling of low entropy gas have been proposed to explain the deviation from self-similarity in the cluster L_x-T_x relation and the observed entropy floor in these systems. However, severe overcooling of gas in groups is necessary for radiative cooling alone to explain the observations. Non-gravitational entropy injection must therefore still be important in these systems. We point out that on scales of groups and below, gas heated to the required entropy floor cannot cool in a Hubble time, regardless of its subsequent adiabatic compression. Preheating therefore shuts off the gas supply to galaxies, and should be an important global feedback mechanism for galaxy formation. Constraints on global gas cooling can be placed from the joint evolution of the comoving star formation rate and neutral gas density. Preheating at high redshift can be ruled out; however the data does not rule out passive gas consumption without inflow since z~2. Since for preheated gas t_cool > t_dyn, we speculate that preheating could play a role in determining the Hubble sequence: at a given mass scale, high sigma peaks in the density field collapse early to form ellipticals, while low sigma peaks collapse late and quiescently accrete preheated gas to form spirals. The entropy produced by large scale shock-heating of the intergalatic medium is significant only at late times, z<1, and cannot produce these effects.Comment: 10 pages, submitted to MNRA

Some properties on GG-evaluation and its applications to GG-martingale decomposition

In this article, a sublinear expectation induced by $G$-expectation is introduced, which is called $G$-evaluation for convenience. As an application, we prove that any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ the decomposition theorem holds and any $\beta>1$ integrable symmetric $G$-martingale can be represented as an It$\hat{o}'s$ integral w.r.t $G$-Brownian motion. As a byproduct, we prove a regular property for $G$-martingale: Any $G$-martingale $\{M_t\}$ has a quasi-continuous versionComment: 22 page

Reionization Through the Lens of Percolation Theory

The reionization of intergalactic hydrogen has received intense theoretical scrutiny over the past two decades. Here, we approach the process formally as a percolation process and phase transition. Using semi-numeric simulations, we demonstrate that an infinitely-large ionized region abruptly appears at an ionized fraction of ~0.1 and quickly grows to encompass most of the ionized gas: by an ionized fraction of 0.3, nearly ninety percent of the ionized material is part of this region. Throughout most of reionization, nearly all of the intergalactic medium is divided into just two regions, one ionized and one neutral, and both infinite in extent. We also show that the discrete ionized regions that exist before and near this transition point follow a near-power law distribution in volume, with equal contributions to the total filling factor per logarithmic interval in size up to a sharp cutoff in volume. These qualities are generic to percolation processes, with the detailed behavior a result of long-range correlations in the underlying density field. These insights will be crucial to understanding the distribution of ionized and neutral gas during reionization and provide precise meaning to the intuitive description of reionization as an "overlap" process.Comment: 16 pages, version accepted by MNRAS (conclusions unchanged from original