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We examine a simple repeated principal-agent model with discounting. There are a risk averse borrower with an unobservable random income and a risk neutral lender. The efficient contract is characterized. It tends to the first-best (constant consumption) contract as the discount factor tends to one and the time horizon extends to infinity. If the time(More)
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)
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)
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)
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)
Let (X t) t≥0 be a continuous-time irreducible Markov chain on a finite statespace E, let v be a map v : E → R\{0} and let (ϕ t) t≥0 be an additive functional defined by ϕ t = t 0 v(X s)ds. We consider the cases where the process (ϕ t) t≥0 is oscillating and where (ϕ t) t≥0 has a negative drift. In each of the cases we condition the process (X t , ϕ t) t≥0(More)