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.

在本文中，我们使用单调算子技术研究了一个完全非线性的演化汉密尔顿-雅各比-贝尔曼（HJB）抛物方程。我们考虑由投资组合优化选择引起的HJB方程，其目标是使投资组合的最终效用的条件期望值最大化。使用所谓的Riccati变换方法，将完全非线性的HJB方程转换为准线性抛物线方程。可以将转换后的抛物线方程视为带源项的方程的多孔介质类型。在某些假设下，我们得到了拟线性抛物方程的扩散函数是全局Lipschitz连续的，这是解决Cauchy问题的关键条件。我们采用Banach不动点定理，在抽象的环境中，在合适的Sobolev空间中，获得变换抛物方程的一般形式的解的存在性和唯一性。一维空间中展示了提出的结果的一些财务应用。