Some remarks on the optional decomposition theorem

摘要: Let S be a vector-valued semimartingale and Z(S) the set of all strictly positive local martingales Z with Z0=1 such that ZS is martingale. Assume V (resp. U) nonnegative process for each Z∈Z(S) ZV supermartingale ZU submartingale sup Z∈z(s),τ∈τ1 E(ZτUτ) < + ∞ where T f denotes finite stopping times). Then admits decomposition V=V0+ϕ·S−C U=U0+ψ·S+A) C A are adapted increasing processes C0=A0=0. The first result slight generalization optional theorem (see [2,4,7]) second one new. As an application to mathematical finance, if interpreted as discounted price stocks, we show contains exactly element iff market complete.