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… (More)
We discuss three forms of convergence in distribution which are stronger than the normal weak convergence. They have the advantage that they are non-topological in nature and are inherited by discontinuous functions of the original random variables—clearly an improvement on 'normal' weak convergence. We give necessary and sufficient conditions for the three… (More)
Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z n 2. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.
In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.
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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… (More)
We consider a model for the control of a satellite—fuel is expended in a linear fashion to move a satellite following a diffusion—the aim is to keep the satellite above a critical level. Under suitable assumptions on the drift and diffusion coefficients , it is shown that the probability of the satellite falling below the critical level is minimised by a… (More)
In this paper, we consider trading with proportional transaction costs as in Schachermayer's paper of 2004. We give a necessary and sufficient condition for A, the cone of claims attainable from zero endowment, to be closed. Then we show how to define a revised set of trading prices in such a way that firstly, the corresponding cone of claims attainable for… (More)
In this paper we give two examples of evanescent Markov chains which exhibit unusual behaviour on conditioning to survive for large times. In the first example we show that the conditioned processes converge vaguely in the discrete topology to a limit with a finite lifetime, but converge weakly in the Martin topology to a non-Markovian limit. In the second… (More)
J o u r n a l o f P r o b a b i l i t y Electron. Abstract This paper describes two explicit couplings of standard Brownian motions B and V , which naturally extend the mirror coupling and the synchronous coupling and respectively maximise and minimise (uniformly over all time horizons) the coupling time and the tracking error of two regime-switching… (More)