J. Appl. Prob. 43, 984–996 (2006) Printed in Israel  Applied Probability Trust 2006 DISCRETIONARY STOPPING OF ONE-DIMENSIONAL ITÔ DIFFUSIONS WITH A STAIRCASE REWARD FUNCTION ANNE LAURE BRONSTEIN, ∗ LANE P. HUGHSTON, ∗∗ ∗∗∗ MARTIJN R. PISTORIUS ∗∗ ∗∗∗∗ and MIHAIL ZERVOS, ∗∗∗∗∗ King’s College London Abstract We consider the problem of optimally stopping a general one-dimensional Itô diffusion X. In particular, we solve the problem that aims at maximising the performance criterion E x [exp(-  τ 0 r(X s ) ds)f (X τ )] over all stopping times τ , where the reward function f can take only a finite number of values and has a ‘staircase’ form. This problem is partly motivated by applications to financial asset pricing. Our results are of an explicit analytic nature and completely characterise the optimal stopping time. Also, it turns out that the problem’s value function is not C 1 , which is due to the fact that the reward function f is not continuous. Keywords: Optimal stopping; American option; principle of smooth fit; local time 2000 Mathematics Subject Classification: Primary 60G40; 93E03; 93E20 Secondary 62P05 1. Introduction This paper is concerned with the problem of optimally stopping the one-dimensional Itô diffusion dX t = b(X t ) dt + σ (X t ) dW t , X 0 = x> 0. (1) Here, W is a standard one-dimensional Brownian motion, and b and σ are deterministic functions such that (1) has a unique weak solution that is nonexplosive and assumes values in the interval (0, ∞). The objective of the discretionary stopping problem is to maximise the performance criterion E x  exp  -  τ 0 r(X s ) ds  f (X τ )  over all stopping times τ , where r> 0 is a given deterministic function. The reward function f takes finite values, and is increasing and piecewise constant, so its graph looks like a staircase with a finite number of steps. Received 26 July 2005; revision received 11 September 2006. ∗ Current address: Laboratoire de Probabilités et Modèles Aléotoires, Université Paris 6, 175 rue du Chevaleret, Paris, 75013, France. Email address: albronstein@hotmail.com ∗∗ Postal address: Department of Mathematics, King’s College London, The Strand, London WC2R 2LS, UK. ∗∗∗ Email address: lane.hughston@kcl.ac.uk ∗∗∗∗ Email address: martijn.pistorius@kcl.ac.uk ∗∗∗∗∗ Current address: Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: m.zervos@lse.ac.uk 984 available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0021900200002370 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 20 Aug 2017 at 02:04:48, subject to the Cambridge Core terms of use,