On the Continuity of Stochastic Exit Time Control Problems

We determine a weaker sufficient condition than that of Theorem 5.2.1 in Fleming and Soner (2006) for the continuity of the value functions of stochastic exit time control problems.Comment: The proof of Lemma 3.1 is slightly modified, and Remark 4.1 is rephrased for better presentations. In addition, some typos are corrected

Outperforming the market portfolio with a given probability

Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show that this value function is the smallest nonnegative viscosity supersolution of a nonlinear PDE. As in Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.], we do not assume the existence of an equivalent local martingale measure, but merely the existence of a local martingale deflator.Comment: Published in at http://dx.doi.org/10.1214/11-AAP799 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org