Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^{0}-$convex topology and in particular a characterization for a locally $L^{0}-$convex module to be $L^{0}-$pre$-$barreled. Section 7 gives some basic results on $L^{0}-$convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty}-$type of conditional convex risk measure and every continuous $L^{p}-$type of convex conditional risk measure ($1\leq p<+\infty$) can be extended to an $L^{\infty}_{\cal F}({\cal E})-$type of $\sigma_{\epsilon,\lambda}(L^{\infty}_{\cal F}({\cal E}), L^{1}_{\cal F}({\cal E}))-$lower semicontinuous conditional convex risk measure and an $L^{p}_{\cal F}({\cal E})-$type of ${\cal T}_{\epsilon,\lambda}-$continuous conditional convex risk measure ($1\leq p<+\infty$), respectively.Comment: 37 page

QCD Factorization in BB Decays into ρπ\rho \pi

Based on the QCD factorization approach we analyse the branching ratios for the channel $B \to \rho \pi$. From the comparisons with experimental data provided by CLEO, BELLE and BABAR we constrain the form factor $F^{B \to \pi}(m_{\rho}^{2})$ and propose boundaries for this form factor depending on the CKM matrix element parameters $\rho$ and $\eta$.Comment: 11 pages, 9 figures. Talk presented at Fourth Tropical Workshop, Cairns, Australia, 9--13 June 2003. Proceedings to be published by AI

On the Cauchy problem for the magnetic Zakharov system

In this paper, we study the Cauchy problem of the magnetic type Zakharov system which describes the pondermotive force and magnetic field generation effects resulting from the non-linear interaction between plasma-wave and particles. By using the energy method to derive a priori bounds and an approximation argument for the construction of solutions, we obtain local existence and uniqueness results for the magnetic Zakharov system in the case of $d=2,3$