On a free boundary problem for an American put option under the CEV process

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free bondary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho$ = $2r/ \sigma^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.Comment: 14 pages, 0 figure

Microscopic structures and thermal stability of black holes conformally coupled to scalar fields in five dimensions

Completely from the thermodynamic point of view, we explore the microscopic character of a hairy black hole of Einstein's theory conformally coupled to a scalar field in five dimensions by means of the Ruppeiner thermodynamic geometry. We demonstrate that the scalar hairy black hole has rich microscopic structures in different parameter spaces. Moreover, we analyze the thermal stability of this black hole in detail.Comment: v1: 15 pages, 1 figure; v2: 19 pages, clarifications and references added; v3: clarifications and references added; v4: 21 pages, clarifications added; v5: minor clarifications and one reference added, final version to appear in Nuclear Physics

Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data

We consider the defocusing, $\dot{H}^1$-critical Hartree equation for the radial data in all dimensions $(n\geq 5)$. We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term $\displaystyle - \int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}\Delta \Big(\frac{1}{|x|}\Big)dxdt$ in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.Comment: 23 pages, 1 figur