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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. [ABSTRACT FROM AUTHOR]

We prove that a fork-convex family $$ \mathbb{W} $$ of nonnegative stochastic processes has an equivalent supermartingale density if and only if the set H of nonnegative random variables majorized by the values of elements of $$ \mathbb{W} $$ at fixed instants of time is bounded in probability. A securities market model with arbitrarily many main risky assets, specified by the set $$ \mathbb{W}\left( \mathbb{S} \right) $$ of nonnegative stochastic integrals with respect to finite collections of semimartingales from an arbitrary indexed family S, satisfies the assumptions of this theorem. [ABSTRACT FROM AUTHOR]

We study the existence of the numéraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numéraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numéraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free-lunch-type notion. In particular, the full strength of the “No Free Lunch with Vanishing Risk” (NFLVR) condition is not needed, only the weaker “No Unbounded Profit with Bounded Risk” (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks. [ABSTRACT FROM AUTHOR]