Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T04:21:08.288Z Has data issue: false hasContentIssue false
  • English
  • Français

A remark on optimal variance stopping problems

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  30 March 2016

Bruno Buonaguidi
Bruno Buonaguidi*
Affiliation:
Bocconi University
*
∗Postal address: Department of Decision Sciences, Bocconi University, Via Roentgen 1, 20136 Milan, Italy. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In an optimal variance stopping problem the goal is to determine the stopping time at which the variance of a sequentially observed stochastic process is maximized. A solution method for such a problem has been recently provided by Pedersen (2011). Using the methodology developed by Pedersen and Peskir (2012), our aim is to show that the solution to the initial problem can be equivalently obtained by constraining the variance stopping problem to the expected size of the stopped process and then by maximizing the solution to the latter problem over all the admissible constraints. An application to a diffusion process used for modeling the dynamics of interest rates illustrates the proposed technique.

Keywords

Constrained and unconstrained variance optimal stopping problemsdiffusion processinterest ratenonlinear optimal stopping problem

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping 90C20: Quadratic programming
Type
Short Communications
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1187 - 1194
Copyright
Copyright © Applied Probability Trust 2015 

References

Ahn, D.-H. and Gao, B. (1999). A parametric nonlinear model of term structure dynamics. Rev. Financial Studies 12, 721–762.CrossRefGoogle Scholar
Aït-Sahalia, Y. (1996). Testing continuous-time models of the spot interest rate. Rev. Financial Studies 9, 385–426.CrossRefGoogle Scholar
Conley, T. G., Hansen, L. P., Luttmer, E. G. J. and Scheinkman, J. A. (1997). Short-term interest rates as subordinated diffusions. Rev. Financial Studies 10, 525–577.Google Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London.Google Scholar
Pedersen, J. L. (2011). Explicit solutions to some optimal variance stopping problems. Stochastics 83, 505–518.Google Scholar
Pedersen, J. L. and Peskir, G. (2012). Optimal mean-variance selling strategies. To appear in Math. Financ. Econ. Google Scholar
Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Shiryaev, A. N. (1978). Optimal Stopping Rules. Springer, New York.Google Scholar