Fundamental Theorems of Asset Pricing for Piecewise Semimartingales of Stochastic Dimension

Abstract

This paper has two purposes. The first is to extend the notions of an n-dimensional semimartingale and its stochastic integral to a piecewise semimartingale of stochastic dimension. The properties of the former carry over largely intact to the latter, avoiding some of the pitfalls of infinite-dimensional stochastic integration. The second purpose is to extend two fundamental theorems of asset pricing (FTAPs): the equivalence of no free lunch with vanishing risk to the existence of an equivalent sigma-martingale measure for the price process, and the equivalence of no arbitrage of the first kind to the existence of an equivalent local martingale deflator for the set of nonnegative wealth processes.

Utility Maximization in a Large Market

“…In contrast to Kabanov and Kramkov (, ), Björk and Näslund () assumed that a large market consists of one probability space, but the number of traded assets is countable, and among other contributions developed the arbitrage pricing theory results in such settings. Note that the models with countably many assets embrace the ones with the stochastic dimension of the stock price process (considered, e.g., in Strong ). De Donno et al.…”