TY - JOUR
T1 - Optimal consumption with reference to past spending maximum
AU - Deng, Shuoqing
AU - Li, Xun
AU - Pham, Huyên
AU - Yu, Xiang
N1 - Funding Information:
We thank two anonymous referees for their helpful comments on the presentation of this paper. H. Pham and X. Yu appreciate the financial support by the PROCORE-France/Hong Kong Joint Research Scheme under no. F-PolyU501/17. X. Yu is partially supported by the Hong Kong Early Career Scheme under grant no. 25302116. X. Li is partially supported by the Hong Kong General Research Fund under grants no. 15213218 and no. 15215319.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2022/4
Y1 - 2022/4
N2 - This paper studies the infinite-horizon optimal consumption problem with a path-dependent reference under exponential utility. The performance is measured by the difference between the nonnegative consumption rate and a fraction of the historical consumption maximum. The consumption running maximum process is chosen as an auxiliary state process, and hence the value function depends on two state variables. The Hamilton–Jacobi–Bellman (HJB) equation can be heuristically expressed in a piecewise manner across different regions to take into account all constraints. By employing the dual transform and smooth-fit principle, some thresholds of the wealth variable are derived such that a classical solution to the HJB equation and feedback optimal investment and consumption strategies can be obtained in closed form in each region. A complete proof of the verification theorem is provided, and numerical examples are presented to illustrate some financial implications.
AB - This paper studies the infinite-horizon optimal consumption problem with a path-dependent reference under exponential utility. The performance is measured by the difference between the nonnegative consumption rate and a fraction of the historical consumption maximum. The consumption running maximum process is chosen as an auxiliary state process, and hence the value function depends on two state variables. The Hamilton–Jacobi–Bellman (HJB) equation can be heuristically expressed in a piecewise manner across different regions to take into account all constraints. By employing the dual transform and smooth-fit principle, some thresholds of the wealth variable are derived such that a classical solution to the HJB equation and feedback optimal investment and consumption strategies can be obtained in closed form in each region. A complete proof of the verification theorem is provided, and numerical examples are presented to illustrate some financial implications.
KW - Consumption running maximum
KW - Exponential utility
KW - Path-dependent reference
KW - Piecewise feedback control
KW - Verification theorem
UR - http://www.scopus.com/inward/record.url?scp=85125875689&partnerID=8YFLogxK
U2 - 10.1007/s00780-022-00475-w
DO - 10.1007/s00780-022-00475-w
M3 - Journal article
AN - SCOPUS:85125875689
SN - 0949-2984
VL - 26
SP - 217
EP - 266
JO - Finance and Stochastics
JF - Finance and Stochastics
IS - 2
ER -