Unifying constructions of martingales associated with processes increasing in the convex order, via Lévy and Sato sheets
The article presents a unified construction of martingales associated to certain processes that are increasing in the convex order (PCOCs, as the authors call them, derived from the French ``processus croissant pour l'ordre convexe''). A family of random variables \((X_t: t \geq 0)\) is called a PCOC if, for all \(t\), \[ E(|X_t|) < \infty \] and if, for any real-valued convex function \(g\), \[ t \mapsto E(g(X_t)) \in (-\infty, \infty] \] is an increasing function. By Jensen's inequality, it is immediately seen that the one-dimensional marginals \(X_t\) of a martingale \(X\) form a PCOC. A deep result by Kellerer states that, for any PCOC \(X\), there exists a martingale \(M\), such that for all \(t\), \(X_t\) and \(M_t\) have the same distribution. However, in general, it is not known how to construct the martingale \(M\). In a collection of previous articles, the authors gave explicit constructions for a number of examples, based on the Brownian sheet, more generally, on Lévy sheets and Sato sheets. In this article, the previous constructions for Lévy sheets and Sato sheets are unified in a general framework, based on space-time harmonic functions of a Markov process.
scientific article; zbMATH DE number 5817634