On the existence of an equivalent supermartingale density for a fork-convex family of stochastic processes

AbstractWe prove that a fork-convex family $$ \mathbb{W} $$ of nonnegative stochastic processes has an equivalent supermartingale density if and only if the setH 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.