Free boundary problem for an optimal investment problem with a borrowing constraint

Chonghu Guan; Xun Li; Rui Zhou; Wenxin Zhou

doi:10.3934/jimo.2021049

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2022, Volume 18, Issue 3: 1915-1934. Doi: 10.3934/jimo.2021049 open access

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Free boundary problem for an optimal investment problem with a borrowing constraint

1.
School of Mathematics, Jiaying University, Meizhou 514015, China
2.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

^* Corresponding author: Rui Zhou
^* Corresponding author: Rui Zhou

Received: June 2020

Revised: December 2020

Published: May 2022

The work is supported by NNSF of China (No.11901244 and No.11901093), Universities and Colleges Special Innovation Project of Guangdong Province (No.2019KTSCX166), and Research Grants Council of Hong kong under grant 15213218 and 15215319

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Abstract

This paper considers an optimal investment problem under CRRA utility with a borrowing constraint. We formulate it into a free boundary problem consisting of a fully nonlinear equation and a linear equation. We prove the existence and uniqueness of the classical solution and present the condition for the existence of the free boundary under a linear constraint on a borrowing rate. Furthermore, we prove that the free boundary is continuous and smooth when the relative risk aversion coefficient is sufficiently small.

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Mathematics Subject Classification: Primary: 91B70, 91G10; Secondary: 35R35, 35B65.

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Figure 1. Free boundaries with various $ k $ and $ b $

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