We consider a nondominated model of a discrete-time financial market where
stocks are traded dynamically, and options are available for static hedging. In
a general measure-theoretic setting, we show that absence of arbitrage in a
quasi-sure sense is equivalent to the existence of a suitable family of
martingale measures. In the arbitrage-free case, we show that optimal
superhedging strategies exist for general contingent claims, and that the
minimal superhedging price is given by the supremum over the martingale
measures. Moreover, we obtain a nondominated version of the Optional
Decomposition Theorem.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1011 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
We prove a weak version of the dynamic programming principle for standard stochastic control problems and mixed control-stopping problems, which avoids the technical difficulties related to the measurable selection argument. In the Markov case, our result is tailor-maid for the derivation of the dynamic programming equation in the sense of viscosity solutions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.