We show that the problem of pricing the American put is equivalent to solving an optimal stopping problem. The optimal stopping problem gives rise to a parabolic free-boundary problem. We show there is a unique solution to this problem which has a lower boundary. We identify an integral equation solved by the boundary and show that it is the unique solution to this equation satisfying certain, natural, additional conditions. The proofs also give a natural decomposition of the price of the American option as the sum of the price of the European option and an 'American premium'.This paper concerns itself with the problem of pricing the basic American put option. The option confers the right to sell a unit of stock at any time up to the time horizon T . We assume that the stock pays no dividends during the lifetime of the option and that the stock price at time t, X t , is an exponential Brownian motion:where (B t ; t ≥ 0) is canonical Brownian motion. Thus X t is the unique (strong) solution to the stochastic differential equation:with fixed initial value X 0 . We also assume that cash generates interest at a fixed rate α(> 0). This paper establishes, using elementary techniques, the 'well-known' result that the fair price for the option, regarded as a function of the present stock price and the time horizon, is the (essentially) unique solution to a parabolic free-boundary problem; Theorem 4.2.1 (see, for example, McKean (1965) and Van Moerbeke (1976). We establish certain basic properties of the (lower) free boundary b(t); and show that b(t) is the unique left-continuous solution of a convolution-type integral equation; Theorem 4.2.2.

We establish necessary and sufficient conditions for an H 1 -martingale to be representable with respect to a collection, X , of local martingales. M ∈ H 1 (P ) is representable if and only if M is a local martingale under all p.m.s Q which are 'uniformly equivalent' to P and which make all the elements of X local martingales (Theorem 1). We then give necessary and sufficient conditions which are easier to verify, and only involve expectations (Theorem 2). We go on to apply these results to the problem of pricing claims in an incomplete financial market-establishing two conjectures of Harrison and Pliska (1981).Key Words: martingale, martingale representation, local martingale, incomplete financial market, contingent claim, attainable claim, fair price. §1. Introduction Harrison and Pliska (1981) showed that every contingent claim in a financial market is attainable (hedgeable) if and only if the collection of martingale measures is a singleton (see Harrison and Pliska (1981) for terminology and assumptions); such a market is called complete. The following question then arises very naturally-'which claims are attainable in an incomplete financial market?'Harrison and Pliska's proof relied on results in Jacod (1979) and a basic equivalence between questions of this type and questions about martingale representations. Essentially any dynamic hedging strategy corresponds to a (vector-valued) previsible process φ, and the (discounted) value of the corresponding portfolio corresponds to the stochastic integral of φ with respect to the vector of (discounted) security prices. The question of attainability then becomes one of representation-' which contingent claims can be represented as a stochastic integral with respect to the vector of discounted security prices?'-whilst the parallel question in martingale representation theory is:1 I am grateful to Wilfrid Kendall, Robin Reed and Marc Yor for helpful and stimulating discussions on the topics in this paper, and to C Stricker and an anonymous referee both for pointing out errors in a previous versions of this paper and for helping me to improve its presentation.

We consider the problem of conditioning a continuous-time Markov chain (on a countably infinite state space) not to hit an absorbing barrier before time T; and the weak convergence of this conditional process as T → ∞. We prove a characterization of convergence in terms of the distribution of the process at some arbitrary positive time, t, introduce a decay parameter for the time to absorption, give an example where weak convergence fails, and give sufficient conditions for weak convergence in terms of the existence of a quasi-stationary limit, and a recurrence property of the original process.