Journal of the
Korean Mathematical Society JKMS

This paper treats Merton's classical portfolio optimization problem for a market participant who invests in safe assets and risky assets to maximize the expected utility. When the state process is a $d$-dimensional Markov diffusion, this problem is transformed into a problem of solving a Hamilton--Jacobi--Bellman (HJB) equation. The main purpose of this paper is to solve this HJB equation by a deep learning algorithm: the deep Galerkin method, first suggested by J. Sirignano and K. Spiliopoulos. We then apply the algorithm to get the solution to the HJB equation and compare with the result from the finite difference method.

Keywords: Merton problem, optimal investment, deep learning algorithm, deep Galerkin method

MSC numbers: Primary 60J70, 91G60

Supported by: Hyungbin Park was supported by Research Resettlement Fund for the new faculty of Seoul National University, South Korea. In
addition, Hyungbin Park was supported by the National Research Foundation of Korea (NRF) grants funded by the Ministry of Science and ICT (No.2017R1A5A1015626, No. 2018R1C1B5085491 and No. 2021R1C1C1011675) and the Ministry of Education (No. 2019R1A6A1A10073437) through the Basic Science Research Program. Financial support from the Institute for Research in Finance and Economics of Seoul National University is gratefully acknowledged.