Abstract
Let (X,Y) be a mean zero martingale pair, i.e., X and Y possess mean zero and E(YIX)=X a.s.. It has been proved in various ways that (1) there exist stopping times τ on Brownian motion {B(t);t≥0} such that B(τ) is distributed like X and {B(tΛτ); t≥0} is uniformly integrable; and (2) for any such τ there exist stopping times τ′ such that τ≤τ′ a.s., (B(τ), B(τ′)) is distributed like (X,Y), and {B(tΛτ′); t≥0} is uniformly integrable. In other words (to explain the role of uniform integrability), a martingale pair can be embedded in a piece of Brownian motion that is itself a martingale.
We will show that unless Y lives on one or two points, there can exist no stopping time τ′ with {B(tΛτ′); t≥0} uniformly integrable and B(τ′) distributed as Y, such that whenever (X,Y) is a martingale pair there exist τ with τ≤τ′ a.s. and B(τ) distributed as X.
on leave from Tel-Aviv University, 1980/1.
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© 1982 Springer-Verlag
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Meilijson, I. (1982). There exists no ultimate solution to Skorokhod's problem. In: Azéma, J., Yor, M. (eds) Séminaire de Probabilités XVI 1980/81. Lecture Notes in Mathematics, vol 920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092802
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DOI: https://doi.org/10.1007/BFb0092802
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