Finite time-horizon optimal investment and consumption with time-varying subsistence consumption constraints

• Published in: Japan journal of industrial and applied mathematics Vol. 38; no. 1; pp. 353 - 377
• Main Authors: Jeon, Junkee, Kang, Myungjoo, Shin, Yong Hyun
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.02.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Cauchy problems; Computational Mathematics and Numerical Analysis; Consumption; Continuity (mathematics); Integral transforms; Martingales; Mathematics; Mathematics and Statistics; Original Paper; Partial differential equations; Policies; Portfolio selection; Integral transform; 91G10; Time-varying subsistence consumption constraints; Utility maximization; Martingale method; 91G80; Cauchy problem
• ISSN: 0916-7005; 1868-937X
• DOI: 10.1007/s13160-020-00440-0

In this paper we consider a general optimal consumption and portfolio selection problem of a finitely-lived agent whose consumption rate process is subject to time-varying subsistence consumption constraints. That is, her consumption rate should be greater than or equal to some convex, non-decreasing and continuous function of time t . Using martingale duality approach and Feynman–Kac formula, we derive the partial differential equation of the Cauchy problem satisfied by the dual value function. We use the integral transform method for solving this Cauchy problem to obtain the general optimal policies in an explicit form. With constant relative risk aversion and constant absolute risk aversion utility functions we illustrate some numerical results of the optimal policies.