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

Strong laws of large numbers for sub-linear expectations

We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.Comment: 10 page

Reionization by Hard Photons: I. X-rays from the First Star Clusters

Observations of the Ly-alpha forest at z~3 reveal an average metallicity Z~0.01 Z_solar. The high-redshift supernovae that polluted the IGM also accelerated relativistic electrons. Since the energy density of the CMB scales as (1+z)^4, at high redshift these electrons cool via inverse Compton scattering. Thus, the first star clusters emit X-rays. Unlike stellar UV ionizing photons, these X-rays can escape easily from their host galaxies. This has a number of important physical consequences: (i) Due to their large mean free path, these X-rays can quickly establish a universal ionizing background and partially reionize the universe in a gradual, homogeneous fashion. If X-rays formed the dominant ionizing background, the universe would have more closely resembled a single-phase medium, rather than a two-phase medium. (ii) X-rays can reheat the universe to higher temperatures than possible with UV radiation. (iii) X-rays counter the tendency of UV radiation to photo-dissociate H2, an important coolant in the early universe, by promoting gas phase H2 formation. The X-ray production efficiency is calibrated to local observations of starburst galaxies, which imply that ~10% of the supernova energy is converted to X-rays. While direct detection of sources in X-ray emission is difficult, the presence of relativistic electrons at high redshift and thus a minimal level of X-ray emission may be inferred by synchrotron emission observations with the Square Kilometer Array. These sources may constitute a significant fraction of the unresolved hard X-ray background, and can account for both the shape and amplitude of the gamma-ray background. This paper discusses the existence and observability of high-redshift X-ray sources, while a companion paper models the detailed reionization physics and chemistry.Comment: Final version accepted by ApJ. 32 pages, 3 figure

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