Finite horizon portfolio selection problems with stochastic borrowing constraints

Junkee Jeon

doi:10.3934/jimo.2019132

Article Contents

2021, Volume 17, Issue 2: 733-763. Doi: 10.3934/jimo.2019132

This issue Previous Article Loss-averse supply chain decisions with a capital constrained retailer Next Article Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model

Finite horizon portfolio selection problems with stochastic borrowing constraints

Junkee Jeon^,

Department of Applied Mathematics & Institute of Natural Science, Kyung Hee University, Yongin, 17104, Republic of Korea

^* Corresponding author: Junkee Jeon
^* Corresponding author: Junkee Jeon

Received: January 31, 2019

Revised: March 31, 2019

Early access: October 2019

Published: March 2021

The first author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Grant No. NRF-2017R1C1B1001811)

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Abstract

In this paper we investigate the optimal consumption and investment problem with stochastic borrowing constraints for a finitely lived agent. To be specific, she faces a credit limit which is a constant fraction of the present value of her stochastic labor income at each time. By using the martingale approach and transformation into an infinite series of optimal stopping problems which has the same characteristic as finding the optimal exercise time of an American option. We recover the value function by establishing a duality relationship and obtain the integral equation representation solution for the optimal consumption and portfolio strategies. Moreover, we provide some numerical illustrations for optimal consumption and investment policies.

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Mathematics Subject Classification: Primary: 91G10, 91G80; Secondary: 90C15.

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Figure 1. Free boundary $ z^{\star}(t) $. Parameter values are given by $ \mu = 0.05, \sigma = 0.2, r = 0.01, \beta = 0.05, \gamma = 2, \mu_{I} = 0.012, \sigma_{I} = 0.1, \nu = 0.3 \;\;\mbox{and}\; T = 10 $

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Figure 2. Simulated paths of wealth to income ratio $ X^{*}/I $, portfolio to income ratio $ \pi^{*}/I $, consumption to income ratio $ c^{*}/I $, the process $ y^{*}D^{*} $, and the process $ D^{*} $. Parameter values are given by $ \mu = 0.05, \sigma = 0.2, r = 0.01, \beta = 0.05, \gamma = 2, \mu_{I} = 0.012, \sigma_{I} = 0.1, \nu = 0.3 \;\;\mbox{and}\; T = 30 $

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