Optimal Bankruptcy Time and Consumption/Investment Policies on an Infinite Horizon with a Continuous Debt Repayment Until Bankruptcy

Abstract

In this paper we consider the optimization problem of an agent who wants to maximize the total expected discounted utility from consumption over an infinite horizon. The agent is under obligation to pay a debt at a fixed rate until he/she declares bankruptcy. At that point, after paying a fixed cost, the agent will be able to keep a certain fraction of the present wealth, and the debt will be forgiven. The selection of the bankruptcy time is taken to be at the discretion of the agent. The novelty of this paper is that at the time of bankruptcy the wealth process has a discontinuity, and that the agent continues to invest and consume after bankruptcy. We show that the solution of a free boundary problem satisfying some additional conditions is the value function of the above optimization problem. Particular examples such as the logarithmic and the power utility functions will be provided, and in these cases explicit forms will be given for the optimal bankruptcy time, investment and consumption processes.

Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints

“…(See Remark 3.1 of Choi and Koo [2], Remark 3.1 of Choi et al [4] and Proposition 2.4 of Jeanblanc et al [9].) That is, for (7) is given by…”

“…Karatzas and Wang [13] were the first to study a discretionary stopping problem by using a martingale method and Filar et al [7] considered the optimal retirement time as a target hitting time using the Markov decision processes. Jeanblanc et al [9] solved an optimal consumption-portfolio selection problem of an agent who is under obligation to pay a debt until she declares bankruptcy.…”

Optimal Bankruptcy with a Continuous Debt Repayment

“…Recently, Choi et al (2015) provides explicit expressions for the minimum wealth boundary when the economic agent faces market participation constraint. Up to our best knowledge, Jeanblanc et al (2004) is the closest work to ours. However, they apply the dynamic programming principle (DPP) to the optimal bankruptcy problem with no wage income.…”

“…Jeanblanc et al (2004) consider the similar problem but they do not consider the labor income even before bankruptcy. Moreover, they apply the DPP developed in Karatzas et al (1987).…”

Comparative Studies on Cooperative Stochastic Differential Game and Dynamic Sequential Game of Economic Maturity

“…A canonical application of the Verification Theorem of HJB Equation (see, Øksendal, 2003) shows the desired assertion. ■ Some papers such as Karatzas and Wang (2000), Jeanblanc et al (2004), and also the textbook of Øksendal and Sulem (2005) study utility maximization with discretionary stopping. Instead of deriving the optimal stopping time and the optimal controls simultaneously, Theorems 1 and 2 establish optimal consumption for any given stopping time based upon our Computation Algorithm defined in Section 2.…”