Necessary and sufficient conditions for the supermartingale property of a stochastic integral with respect to a local martingale

Abstract

This paper deals with necessary and sufficient conditions for the supermartingale property of a stochastic integral with respect to a local martingale. A basic answer is due to Ansel and Stricker [1].

Recently, Schachermayer [3], and Kabanov and Stricker [2] have also dealt with this problem requiring an integrability condition at arbitrary sequences of stopping times (cf. (7)). The subject of this paper is how to improve these results by imposing this integrability condition at a considerably smaller class of stopping times (cf. (3)). As a result it turns out that it suffices to impose this integrability condition at the time horizon and at one particular sequence of hitting times (cf. Theorem 2). By means of a counterexample (cf. Section 1), it is shown that none of the two conditions can be omitted. As a side result we give an application to mathematical finance (cf. Section 3).

Keywords: stochastic integral, local martingale, supermartingale, submartingale, no-arbitrage

AMS 1991 subject classifications. Primary 60H05; secondary 90A09