Solution of the Optimal Stopping Problem for One-Dimensional Diffusion Based on a Modification of the Payoff Function
A problem of optimal stopping for one-dimensional time-homogeneous regular diffusion with the infinite horizon is considered. The diffusion takes values in a finite or infinite interval ]a,b[. The points a and b may be either natural or absorbing or refle
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Abstract A problem of optimal stopping for one-dimensional time-homogeneous regular diffusion with the infinite horizon is considered. The diffusion takes values in a finite or infinite interval a; bŒ. The points a and b may be either natural or absorbing or reflecting. The diffusion may have a partial reflection at a finite number of points. A discounting and a cost of observation are allowed. Both can depend on the state of the diffusion. The payoff function g.z/ is bounded on any interval Œc; d , where a < c < d < b, and twice differentiable with the exception of a finite (may be empty) set of points, where the functions g.z/ and g 0 .z/ may have a discontinuities of the first kind. Let L be an infinitesimal generator of diffusion which includes the terms corresponding to the discounting and the cost of observation. We assume that the set fz W Lg.z/ > 0g consists of a finite number of intervals. For such problem we propose a procedure of constructing the value function in a finite number of steps. The procedure is based on a fact that on intervals where Lg.z/ > 0 and in neighborhoods of points of partial reflections, points of discontinuities, and points a or b in case of reflection, one can modify the payoff function preserving the value function. Many examples are considered. Keywords Markov chain • Markov process • One-dimensional diffusion • Optimal stopping • Elimination algorithm
Mathematics Subject Classification (2010): 60G40, 60J60, 60J65.
E. Presman () CEMI RAS, 47 Nakhimovsky prospect, Moscow, 117428, Russia e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 22, © Springer-Verlag Berlin Heidelberg 2013
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1 Introduction We consider a time-homogenous strong Markov process Z D .Zt /t 0 with values S in X e, where .X; B/ is a measurable space, and e is an absorbing state. The time may be discrete or continuous. We assume that Z is defined on some filtered probabilistic space and that the following measurable functions are given: .z/ 0 – killing intensity; g.z/ – payoff function, g.e/ D 0; c.z/ – cost of observations, c.e/ D 0. In the continuous time we consider the functional 3 2 Z V .z; / D Ez 4g.Z / c.Zs /ds 5 ; (1) 0
where is an arbitrary stopping time with respect to the given filtration. In the discrete time instead of the integral one has a sum from 0 to 1. We assume that the expectation is defined for any . The optimal stopping problem consists in a maximization of the functional (1). This problem is equivalent to the problem where instead of absorbtion one has in the continuous time the functional 3 2 R Rs Z .Zu /d u .Zu /d u V .z; / D EN z 4g.Z /e 0 ds c.Zs /e 0 ds 5 ; 0
where EN corresponds to the process without absorbtion. In the discrete time instead Rt Q 1 .1 .Zu //. of exp. .Zu /d u/ one has tuD0 0
The aim of this paper is to present a procedure for constructing the value function V .z/ D sup
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