Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-30T21:17:15.171Z Has data issue: false hasContentIssue false
  • English
  • Français

An asymptotic expansion for the optimal stopping boundary in problems with non-linear costs of observation

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  14 July 2016

A. Irle ,
O. Kubillus  and
V. Paulsen
A. Irle*
Affiliation:
Universität Kiel
O. Kubillus*
Affiliation:
Universität Kiel
V. Paulsen*
Affiliation:
Universität Kiel
*
∗ Postal address: Universität Kiel, Mathematisches Seminar, Ludewig Meyn Str. 4, D-24098 Kiel, Germany.
∗ Postal address: Universität Kiel, Mathematisches Seminar, Ludewig Meyn Str. 4, D-24098 Kiel, Germany.
∗ Postal address: Universität Kiel, Mathematisches Seminar, Ludewig Meyn Str. 4, D-24098 Kiel, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

The assumption of linear costs of observation usually leads to optimal stopping boundaries which are straight lines. For non-linear costs of observation, the question arises of how the shape of cost functions influences the shape of optimal stopping boundaries. In Irle (1987), (1988) it was shown that, under suitable assumptions on c, for the problem of optimal stopping (Wt + x)+ - c(s + t), t ∊ [0,∞), the optimal stopping boundary h(t) can be enscribed between k1/c'(t) and k2/c'(t) for some constants k1, k2. In this paper we find the exact asymptotic expansion h(t) = 1/(4c'(t))(1 + o(1)).

Keywords

optimal stoppingnon-linear boundaryasymptotic expansion

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 67 - 79
Copyright
Copyright © by the Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berk, R. H. (1975). Locally most powerful sequential tests. Ann. Statist. 3, 373–381.CrossRefGoogle Scholar
Cannon, J. R. (1984). The One-dimensional Heat Equation (Encycl. Math. Appl. 23). Cambridge University Press.Google Scholar
Chernoff, H. (1968). Optimal stochastic control. Sankhyā A 30, 221–252.Google Scholar
Irle, A. (1980). Locally best tests for Gaussian processes. Metrika 27, 15–28.CrossRefGoogle Scholar
Irle, A. (1987). On optimal stopping with concave costs of observation. Sequential Anal. 6, 1–19.CrossRefGoogle Scholar
Irle, A. (1988). On optimal stopping with non-linear costs of observation. Sequential Anal. 7, 191–202.CrossRefGoogle Scholar
Lai, T. L. (1973). Optimal stopping and sequential tests which minimize the maximum expected sample size. Ann. Statist. 1, 659–673.CrossRefGoogle Scholar
Shiryayev, A. N. (1978). Optimal Stopping Rules. Springer, Berlin.Google Scholar