# SFB 303 Discussion Paper No. B-294

**Author-Name:** Kramkov, D.O.

**Title:** Optional decomposition of supermartingales and hedging
contingent claims in incomplete security markets

**Abstract:** Let M(X) be a family of all equivalent local martingale
measures for some locally bounded d-dimensional process X, and V be a
positive process. Main result of the paper (Theorem 2.1) states that the
process V is a supermartingale whatever Q in M(X), if and only if this
process admits the following decomposition: V_t = V_0 + \int_0^t H_s
dX_s - C_t, t>= 0, where H is an integrand for X, and C is an adapted
increasing process. We call such a representation the optional because,
in contrast to Doob-Meyer decomposition, it generally exists only with
an adapted (optional) process C. We apply this decomposition to the problem
of hedging European and American style contingent claims in a setting
of incomplete security markets.

**Keywords:** Doob-Meyer decomposition, optional decomposition,
martingale measure, stochastic integral, semimartingale topology, incomplete
market, hedging, options

**JEL-Classification-Number:** G13

**Creation-Date:** October 1994

**URL:**
../1994/b/bonnsfb294.pdf
SFB 303 Homepage

17.02.1998, © Webmaster