Transitivity in Problems of Optimal Stopping

Abstract

In a sequential decision problem it is usually assumed that the available information is represented by an increasing family $\mathscr{F}$ of $\sigma$-algebras. Often a reduction, e.g., according to principles of sufficiency or invariance, is performed which yields a smaller family $\mathscr{G}$. The consequences of such a reduction for problems of optimal stopping are treated in this paper. It is shown that $\mathscr{G}$ is transitive for $\mathscr{F}$ (in the Bahadur sense) if and only if for any stochastic process adapted to $\mathscr{G}$ the value (i.e., maximal reward by optimal stopping) under $\mathscr{G}$ and the value under $\mathscr{F}$ are equal.