Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem

• Published in: Japan journal of industrial and applied mathematics Vol. 38; no. 3; pp. 693 - 713
• Main Authors: Udeani, Cyril Izuchukwu, Ševčovič, Daniel
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01-09-2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Cauchy problems; Computational Mathematics and Numerical Analysis; Existence theorems; Fixed points (mathematics); Mathematics; Mathematics and Statistics; Operators (mathematics); Optimization; Original Paper; Porous media; Sobolev space; 70H20; Hamilton–Jacobi–Bellman equation; Dynamic stochastic portfolio optimization; 91B16; 91B70; Maximal monotone operator; 90C15; Riccati transformation; 35K55; 34E05
• ISSN: 0916-7005; 1868-937X
• DOI: 10.1007/s13160-021-00468-w

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.