Time-randomized stopping problems for a family of utility functions

Abstract

This paper studies stopping problems of the form $V=\inf_{0 \leq \tau \leq T} \mathbb{E}[U(\frac{\max_{0\le s \le T} Z_s }{Z_\tau})]$ for strictly concave or convex utility functions U in a family of increasing functions satisfying certain conditions, where Z is a geometric Brownian motion and T is the time of the nth jump of a Poisson process independent of Z. We obtain some properties of $V$ and offer solutions for the optimal strategies to follow. This provides us with a technique to build numerical approximations of stopping boundaries for the fixed terminal time optimal stopping problem presented in [J. Du Toit and G. Peskir, Ann. Appl. Probab., 19 (2009), pp. 983--1014].

Citation

Pérez López, I., & Le, H. (2015). Time-randomized stopping problems for a family of utility functions. SIAM Journal on Control and Optimization, 53(3), https://doi.org/10.1137/130946800

• Journal Article Type: Article
• Acceptance Date: Mar 9, 2015
• Publication Date: Jan 1, 2015
• Deposit Date: Apr 11, 2016
• Publicly Available Date: Apr 11, 2016
• Journal: SIAM Journal on Control and Optimization
• Print ISSN: 0363-0129
• Electronic ISSN: 1095-7138
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 53
• Issue: 3
• DOI: https://doi.org/10.1137/130946800
• Keywords: optimal stopping, randomization, boundary value problem
• Public URL: https://nottingham-repository.worktribe.com/output/992883
• Publisher URL: http://epubs.siam.org/doi/10.1137/130946800