Hamilton-Jacobi-Bellman Equation Arising from Optimal Portfolio Selection Problem

Paper: arXiv 2308.02627

Authors: Daniel Sevcovic, Cyril Izuchukwu Udeani

Abstract

The Hamilton-Jacobi-Bellman equation arising from the optimal portfolio selection problem is studied by means of the maximal monotone operator method. The existence and uniqueness of a solution to the Cauchy problem for the nonlinear parabolic partial integral differential equation in an abstract setting are investigated by using the Banach fixed-point theorem, the Fourier transform, and the monotone operators technique.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical analysis of the Hamilton-Jacobi-Bellman equation using advanced mathematical tools like monotone operators and Fourier transforms. However, it is purely theoretical with no empirical backtesting or real-world data validation included in the provided summary.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve HJB Equation for Optimal Portfolio Selection] --> B[Input Data: Equities & PIDEs Parameters]
    B --> C[Methodology: Setup Cauchy Problem in Abstract Setting]
    C --> D[Apply Fourier Transform & Banach Fixed-Point Theorem]
    D --> E[Computational Process: Maximal Monotone Operator Technique]
    E --> F[Outcomes: Proven Existence & Uniqueness of Solution]