  This paper treats the Merton problem how to invest in safe assets and risky
assets to maximize an investor's utility, given by investment opportunities
modeled by a $d$-dimensional state process. The problem is represented by a
partial differential equation with optimizing term: the Hamilton-Jacobi-Bellman
equation. The main purpose of this paper is to solve partial differential
equations derived from the Hamilton-Jacobi-Bellman equations with a deep
learning algorithm: the Deep Galerkin method, first suggested by Sirignano and
Spiliopoulos (2018). We then apply the algorithm to get the solution of the PDE
based on some model settings and compare with the one from the finite
difference method.
