An analytical solution for the HJB equation arising from the Merton problem

In this paper, an analytical solution for the well-known Hamilton–Jacobi–Bellman (HJB) equation that arises from the Merton problem subject to general utility functions is presented for the first time. The solution presented in this paper is written in the form of a Taylor’s series expansion and constructed through the homotopy analysis method (HAM). The fully nonlinear HJB equation is decomposed into an infinite series of linear PDEs which can be solved analytically. Four examples are presented with the first two cases showing the accuracy of our solution approach; while the last two demonstrating its versatility.