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

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.