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...re replaced by polyhedral concave functions denoted by the same symbols in Algorithm 3.1, the maximum in (1.1) corresponds to taking the concave cap of Vt and Yt, whereas the expectation in (1.2) is replaced by the restriction to the bid-ask spread interval of the concave cap of the functions Zt+1 over all successor nodes. Because of this correspondence, Algorithm 3.1 can be regarded as a natural extension of the Snell envelope construction. Similar results, algorithms and representations have been established for European options in the general discrete setting by Roux, Tokarz and Zastawniak [RTZ06], and for American options under small transaction costs in the binomial tree model by Tokarz and Zastawniak [TZ05]. The case of American options in an arbitrary discrete market model subject to arbitrary transaction costs turns out to be significantly more challenging due to the appearance of mixed stopping times. Nevertheless, the underlying idea is similar to that in the European options paper [RTZ06]. Consider the concave function x 7→ Zxt such that Zxt = inf(αt + xβt), (1.3) the infimum being taken over all portfolios (αt, βt) of cash (or bonds) and stock held at time t that allow to supe...