Convexity of ruin probability and optimal dividend strategies for a general Levy process

In this paper, we consider the optimal dividends problem for a company whose cash reserves follow a general Levy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we appeal to very recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy.Comment: 19 pages, corrected some typo

An Apparent Redshift Dependence of Quasar Continuum: Implication for Cosmic Dust Extinction?

We investigate the luminosity and redshift dependence of the quasar continuum by means of composite spectrum using a large non-BAL radio-quiet quasar sample drawn from the Sloan Digital Sky Survey. Quasar continuum slopes in the UV-Opt band are measured at two different wavelength ranges, i.e., $\alpha_{\nu12}$ ($1000\sim 2000 \rm\AA$) and $\alpha_{\nu24}$ ($2000 \sim 4000 \rm\AA$) derived from power law fitting. Generally, the UV spectra slope becomes harder (higher $\alpha_{\nu}$) towards higher bolometric luminosity. On the other hand, when quasars are further grouped into luminosity bins, we find both $\alpha_{\nu12}$ and $\alpha_{\nu24}$ show significant anti-correlation with redshift (i.e., quasar continuum becomes redder towards higher redshift). We suggest that the cosmic dust extinction is very likely the cause of this observed $\alpha_\nu-z$ relation. We build a simple cosmic dust extinction model to quantify the observed reddening tendency and find an effective dust density $n\sigma_v \sim 10^{-5}h~\rm Mpc^{-1}$ at $z<1.5$. The other possibilities that could produce such a reddening effect have also been discussed.Comment: 6 pages, 5 figures; published in ApJ