  In this paper, we investigate a fully nonlinear evolutionary
Hamilton-Jacobi-Bellman (HJB) parabolic equation utilizing the monotone
operator technique. We consider the HJB equation arising from portfolio
optimization selection, where the goal is to maximize the conditional expected
value of the terminal utility of the portfolio. The fully nonlinear HJB
equation is transformed into a quasilinear parabolic equation using the
so-called Riccati transformation method. The transformed parabolic equation can
be viewed as the porous media type of equation with source term. Under some
assumptions, we obtain that the diffusion function to the quasilinear parabolic
equation is globally Lipschitz continuous, which is a crucial requirement for
solving the Cauchy problem. We employ Banach's fixed point theorem to obtain
the existence and uniqueness of a solution to the general form of the
transformed parabolic equation in a suitable Sobolev space in an abstract
setting. Some financial applications of the proposed result are presented in
one-dimensional space.
