Abstract
The optimal stopping problem is studied in the paper for functions dependent on the absolute maximum of some homogeneous diffusion. The cases of infinite and finite time horizons are considered. A differential equation for the optimal stopping boundary is obtained in both the cases. A principle of maximum is proved for a function satisfying a single-crossing condition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Shiryaev, Z. Xu, and X. Y. Zhou, “Thou Shalt Buy and Hold,” Quant. Finance 8 (8), 765 (2008).
J. du Toit and G. Peskir, “Selling a Stock at the Ultimate Maximum,” Ann. Appl. Probab. 19 (3), 983 (2009).
M. Dai, H. Jin, Y. Zhong, et al., “Buy Low and Sell High,” Contemp. Quant. Finance 1, 317 (2010).
S. E. Graversen, G. Peskir, and A. N. Shiryaev, “Stopping Brownian Motion without Anticipation as Close as Possible to its Ultimate Maximum,” Theory Probab. and Its Appl. 45 (1), 41 (2001).
J. du Toit and G. Peskir, “The Trap of Complacency in Predicting the Maximum,” Ann. Probab. 35 (1), 340 (2007).
J. L. Pedersen, “Optimal Prediction of the Ultimate Maximum of Brownian Motion,” Stochastics and Stochast. Repts. 75 (4), 205 (2003).
G. Peskir, “Optimal Stopping of the Maximum Process: the Maximality Principle,” Ann. Probab. 26 (4), 1614 (1998).
P. Milgrom and I. Segal, “Envelope Theorems for Arbitrary Choice Sets,” Econometrica 70 (2), 583 (2002).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.A. Kamenov. 2015. published in Vestnik Moskovskogo Universiteta. Matematika. Mekhanika. 2015. Vol. 70, No. 5, pp. 7-13.
About this article
Cite this article
Kamenov, A.A. Optimal stopping for absolute maximum of homogeneous diffusion. Moscow Univ. Math. Bull. 70, 202–207 (2015). https://doi.org/10.3103/S0027132215050022
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027132215050022