On the Existence of an Equivalent Supermartingale Density for a Fork-Convex Family of Stochastic Processes

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 $\mathbb W(\mathbb S)$ of nonnegative stochastic integrals with respect to finite collections of semimartingales from an arbitrary indexed family $\mathbb S$, satisfies the assumptions of this theorem.