Duality for optimal consumption with randomly terminating income
File(s)
Accepted version
Author(s)
Davey, Ashley
Monoyios, Michael
Zheng, Harry
Type
Journal Article
Abstract
We establish a rigorous duality theory, under No Unbounded Profit with Bounded Risk, for an infinite horizon problem of optimal consumption in the presence of an income stream that can terminate randomly at an exponentially distributed time, independent of the asset prices. We thus close a duality gap encountered in the Davis-Vellekoop example in a version of this problem in a Black-Scholes market. Many of the classical tenets of duality theory hold, with the notable exception that marginal utility at zero initial wealth is finite. We use as dual variables a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption in excess of income is a supermartingale. We show that the space of discounted local martingale deflators is dense in our dual domain, so that the dual problem can also be expressed as an infimum over the discounted local martingale deflators. We characterize the optimal wealth process, showing that optimal deflated wealth is a potential decaying to zero, while deflated wealth plus cumulative deflated consumption over income is a uniformly integrable martingale at the optimum. We apply the analysis to the Davis-Vellekoop example and give a numerical solution.
Date Issued
2021-10-01
Date Acceptance
2021-05-28
Citation
Mathematical Finance, 2021, 31 (4), pp.1275-1314
ISSN
0960-1627
Publisher
Wiley
Start Page
1275
End Page
1314
Journal / Book Title
Mathematical Finance
Volume
31
Issue
4
Copyright Statement
© 2021 Wiley Periodicals LLC. This is the peer reviewed version of the following article, which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1111/mafi.12322. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/V008331/1
Subjects
Social Sciences
Science & Technology
Physical Sciences
Business, Finance
Economics
Mathematics, Interdisciplinary Applications
Social Sciences, Mathematical Methods
Business & Economics
Mathematics
Mathematical Methods In Social Sciences
duality
HJB equation
portfolio optimization
supermartingale deflator
terminating income
utility from consumption
OPTIMAL INVESTMENT
SUFFICIENT CONDITIONS
BIPOLAR THEOREM
VERSION
Finance
0102 Applied Mathematics
1502 Banking, Finance and Investment
Publication Status
Published
Date Publish Online
2021-06-15