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Some remarks on the optional decomposition theorem

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1686)

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.

Keywords

  • Process Versus
  • Contingent Claim
  • Security Market
  • Local Martingale
  • Predictable Process

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References

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© 1998 Springer-Verlag

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Stricker, C., Yan, J.A. (1998). Some remarks on the optional decomposition theorem. In: Azéma, J., Yor, M., Émery, M., Ledoux, M. (eds) Séminaire de Probabilités XXXII. Lecture Notes in Mathematics, vol 1686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101750

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  • DOI: https://doi.org/10.1007/BFb0101750

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