Dominated families of martingale, supermartingale and quasimartingale laws
The paper deals with two types of tamed (i.e. absolutely continuous with respect to its selected element) sets of probabilities on a probability space \((\Omega ,\mathcal F,P)\). The first type is formed by sets of probabilities \(Q << P\) such that a given convex set \(K\) of \(P\)-a.s. finite random variables (space \(L^{0}(\Omega ,\mathcal F,P)\)) has a uniformly finite \(Q\)-expectation. Such a set of probabilities contains an element equivalent to \(P\) if and only if a certain spanning condition (C) in the space \(L^{0}(\Omega ,\mathcal F,P)\) is satisfied. The second type contains probability sets which consist of quasi-, semi-, {super-,} sub- or simply martingale laws for an at most countable family \(\chi \) of adapted stochastic processes, i.e. sets of probabilities under which all members of \(\chi \) are quasi-, \dots , -martingales. It is shown that the second type set is also of the first type, provided that \(K\) is generated by stochastic integrals over \(\chi \), the integrands being elementary predictable processes with an additional property. Thus, a characterization of quasi-, \dots , -martingales is obtained, with an application to securities markets with frictions. An equivalent quasi-, \dots , -martingale measure exists on such a market if and only if there are no extreme arbitrage opportunities, a condition equivalent to (C).
scientific article; zbMATH DE number 1226253