A Functional Approach to FBSDEs and Its Application in Optimal Portfolios

In Liang et al (2009), the current authors demonstrated that BSDEs can be reformulated as functional differential equations, and as an application, they solved BSDEs on general filtered probability spaces. In this paper the authors continue the study of functional differential equations and demonstrate how such approach can be used to solve FBSDEs. By this approach the equations can be solved in one direction altogether rather than in a forward and backward way. The solutions of FBSDEs are then employed to construct the weak solutions to a class of BSDE systems (not necessarily scalar) with quadratic growth, by a nonlinear version of Girsanov's transformation. As the solving procedure is constructive, the authors not only obtain the existence and uniqueness theorem, but also really work out the solutions to such class of BSDE systems with quadratic growth. Finally an optimal portfolio problem in incomplete markets is solved based on the functional differential equation approach and the nonlinear Girsanov's transformation.Comment: 26 page

The Luminosity - E_p Relation within Gamma--Ray Bursts and Implications for Fireball Models

Using a sample of 2408 time-resolved spectra for 91 BATSE gamma-ray bursts (GRBs) presented by Preece et al., we show that the relation between the isotropic-equivalent luminosity (L_iso) and the spectral peak energy (E_p) in the cosmological rest frame, L_iso \propto E_p^2, not only holds within these bursts, but also holds among these GRBs, assuming that the burst rate as a function of redshift is proportional to the star formation rate. The possible implications of this relation for the emission models of GRBs are discussed. We suggest that both the kinetic-energy-dominated internal shock model and the magnetic-dissipation-dominated external shock model can well interpret this relation. We constrain the parameters for these two models, and find that they are in a good agreement with the parameters from the fittings to the afterglow data (abridged).Comment: 3 pages plus 5 figures, emulateapj style, accepted for publication in ApJ Letter

Constraining ΩM\Omega_M and Dark Energy with Gamma-Ray Bursts

An $E_{\gamma,{\rm jet}}\propto {E'_p}^{1.5}$ relationship with a small scatter for current $\gamma$-ray burst (GRB) data was recently reported, where $E_{\gamma,{\rm jet}}$ is the beaming-corrected $\gamma$-ray energy and $E'_p$ is the $\nu F_\nu$ peak energy in the local observer frame. By considering this relationship for a sample of 12 GRBs with known redshift, peak energy, and break time of afterglow light curves, we constrain the mass density of the universe and the nature of dark energy. We find that the mass density $\Omega_M=0.35\pm^{0.15}_{0.15}$ (at the $1\sigma$ confident level) for a flat universe with a cosmological constant, and the $w$ parameter of an assumed static dark-energy equation of state $w=-0.84\pm^{0.57}_{0.83}$ ($1\sigma$). Our results are consistent with those from type Ia supernovae. A larger sample established by the upcoming {\em Swift} satellite is expected to provide further constraints.Comment: 8 pages including 4 figures, to appear in ApJ Letters, typos correcte

The Peak Energy Distribution of the nu F_nu Spectra and the Implications for the Jet Structure Models of Gamma-Ray Bursts

We study the peak energy ($E_{\rm{p}}$) distribution of the $\nu F_{\nu}$ spectra of gamma-ray bursts (GRBs) and X-ray flashes (XRFs) with a sample of 57 bursts observed by {\em High Energy Transient Explorer 2} ({\em HETE-2}) French Gamma Telescope and discuss its implications for the jet structure models. Combining the observed $E_{\rm{p}}$ distribution of {\em HETE--2} GRBs/XRFs with that of BATSE GRBs, we find that the observed $E_{\rm{p}}$ distribution of GRBs/XRFs is a bimodal one with peaks of $\lesssim 30$ keV and $\sim 160-250$ keV. According to the recently-discovered equivalent-isotropic energy-$E_{\rm{p}}$ relationship, such a bimodal distribution implies a two-component structure of GRB/XRF jets. A simple simulation analysis shows that this structured jet model does roughly reproduce a bimodal distribution with peaks of $\sim 15$ and $\sim 200$ keV. We argue that future observations of the peak of $\sim 15$ keV in the $E_{\rm{p}}$ distribution would be evidence supporting this model. {\em Swift}, which covers an energy band of 0.2--150 keV, is expected to provide a key test for our results.Comment: 4 pages plus 2 figures, emulateapj style, ApJ Letters, 608, in pres

Linear scaling calculation of maximally-localized Wannier functions with atomic basis set

We have developed a linear scaling algorithm for calculating maximally-localized Wannier functions (MLWFs) using atomic orbital basis. An O(N) ground state calculation is carried out to get the density matrix (DM). Through a projection of the DM onto atomic orbitals and a subsequent O(N) orthogonalization, we obtain initial orthogonal localized orbitals. These orbitals can be maximally localized in linear scaling by simple Jacobi sweeps. Our O(N) method is validated by applying it to water molecule and wurtzite ZnO. The linear scaling behavior of the new method is demonstrated by computing the MLWFs of boron nitride nanotubes.Comment: J. Chem. Phys. in press (2006