Consistent price systems under model uncertainty
The authors give a version of the fundamental theorem of asset pricing for discrete-time models with transaction costs and uncertainty. The investment opportunities in the model are described by a set-valued stochastic process. For each trading day there is given a solvency cone \(K_t\) -- a random set of positions that can be turned into nonnegative ones by immediate exchange (accounting for transaction costs). The \(K_t\) is a convex cone generated by a finite number of vectors. The set \(K^\ast_t\backslash\{0\}\) is interpreted as the set of possible frictionless prices for assets. For each moment, there is given a nonempty convex set of probability measures \(\mathcal{P}_t\) that represents the model uncertainty. The authors define two properties of the model. The first one, \(\text{NA}_2(\mathcal{P})\), is an augmentation of the no-arbitrage condition. It states that \(\zeta\in K_{t+1}\) implies \(\zeta\in K_t\) (\(\mathcal{P}\)-quasi surely). The second one, PCE (price system is extendable), states that for each measurable random variable \(Y\) taking values in \(\operatorname{int} K^\ast_t\) there exists a probability measure \(Q\) and a stochastic process \((Z_s)_{s=t,\dots,T}\) starting as \(Y\), such that \(Z_s\in K^\ast_s\) and \(Z_s\) is a \(Q\)-martingale. The main result of the article states that \(\text{NA}_2(\mathcal{P})\) and PCE are equivalent.
scientific article; zbMATH DE number 6560417
scientific article