On the existence of an equivalent supermartingale density for a fork-convex family of stochastic processes
The author introduces the notion of fork-convex family of stochastic processes that in some sense generalizes the usual notion of convexity. It is proved that a fork-convex family \(\mathbb{W}\) admits 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. This result has a clear interpretation in the framework of the mathematical theory of arbitrage, and what is more, the number of assets can be arbitrary. Namely, the notion of No Unbounded Profit with Bounded Risk (NUPBR) is considered and it is proved that the NUPBR condition is satisfied if and only if there exists an equivalent supermartingale density for the corresponding set of nonnegative stochastic integrals with respect to the semimartingales.
fork-convex family of nonnegative stochastic processes
scientific article; zbMATH DE number 5791084
scientific article