Fundamental Theorem of Asset Pricing, Stochastic Dimension

The purpose of this paper is two-fold. First is to extend the notions of an n-dimensional semimartingaleand its stochastic integral to a piecewise semimartingale of stochastic dimension. The propertiesof the former carry over largely intact to the latter, avoiding some of the pitfalls of infinite-dimensionalstochastic integration. Second is to extend two fundamental theorems of asset pricing (FTAPs): theequivalence of no free lunch with vanishing risk to the existence of an equivalent sigma-martingale measurefor the price process, and the equivalence of no arbitrage of the first kind to the existence of anequivalent local martingale deflator for the set of nonnegative wealth processes.[...]