Some remarks on the optional decomposition theorem

Abstract

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