Optional decomposition of optional supermartingales and applications to filtering and finance

ABSTRACT

The classical Doob–Meyer decomposition and its uniform version the optional decomposition are stated on probability spaces with filtrations satisfying the usual conditions. However, the comprehensive needs of filtering theory and mathematical finance call for their generalizations to more abstract spaces without such technical restrictions. The main result of this paper states that there exists a uniform Doob–Meyer decomposition of optional supermartingales on unusual probability spaces. This paper also demonstrates how this decomposition works in the construction of optimal filters in the very general setting of the filtering problem for optional semimartingales. Finally, the application of these optimal filters of optional semimartingales to mathematical finance is presented.

KEYWORDS:

• Optional decomposition
• uniform Doob–Meyer decomposition
• unusual probability spaces
• optional semimartingales
• filtering of semimartingales
• financial model with incomplete information

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSERC under Grant 5901. Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada.