There Exists No Ultimate Solution to Skorokhod ’ S Problem

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 T on Brownian motion {B(t); such that B(T) is distributed like X and is uniformly integrable; and (2) for any such t there exist stopping times i’ such that T I’ a.s., (B(T), B(I’)) is distributed like (X,Y) and {B(tAT’); 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 ~hat unless Y lives on one or two points, there can exist no stopping time tl with (8(tAT’); uniforr.:-1y integrable and 8(T’) distributed as Y, such that whenever .{ X,Y) is a martingale pair there exist t with a.s. and B(r) distributed as X. * On-leave from Tel-Aviv University, 1980/1.