Martingales on random sets and the strong martingale property. (English) Zbl 0953.60021

Sharpe, Michael J.

Electron. J. Probab. 5, Paper No. 2, 17 p. (2000).

Summary: Let \(X\) be a process defined on an optional random set. The paper develops two different conditions on \(X\) guaranteeing that it is the restriction of a uniformly integrable martingale. In each case, it is supposed that \(X\) is the restriction to \(\Lambda\) of some special semimartingale \(Z\) with canonical decomposition \(Z=M+A\). The first condition, which is both necessary and sufficient, is an absolute continuity condition on \(A\). Under additional hypotheses, the existence of a martingale extension can be characterized by a strong martingale property of \(X\) on \(\Lambda\). Uniqueness of the extension is also considered.

MSC:

60G44

Martingales with continuous parameter

Keywords:

random set; martingale; strong martingale property