On a free boundary problem for an optimal investment problem with different interest rates

This paper discusses an investment problem for a single agent with higher borrowing interest rate than lending in the market. The objective is to maximize the expected discounted utility of terminal wealth by choosing portfolio of one risk asset and the bank account. The objective function is the solution of a free boundary problem with two nonlinear equations and one linear equation. The main contribution is that the existence of free boundary lines is proved in all situations and the design methods can be generally applied to other similar problems.

Keywords

fully nonlinear equation, free boundary lines, stochastic optimal control, optimal investment, different interest rates

2010 Mathematics Subject Classification

35R35, 91B70, 93E20

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The work is supported by NNSF of China (No.11901244, No.11626117, and No.11601163), NSF of Guangdong Province of China (No. 2016A030307008).

Received 3 November 2018

Accepted 3 September 2019

Published 1 April 2020