Paper: arXiv 2507.19824

Authors: Xiaomin Shi, Zuo Quan Xu

Abstract

In this paper, we investigate mean-variance (MV) portfolio selection problems with jumps in a regime-switching financial model. The novelty of our approach lies in allowing not only the market parameters – such as the interest rate, appreciation rate, volatility, and jump intensity – to depend on the market regime, but also in permitting stock prices to experience jumps when the market regime switches, in addition to the usual micro-level jumps. This modeling choice is motivated by empirical observations that stock prices often exhibit sharp declines when the market shifts from a bullish’’ to a bearish’’ regime, and vice versa. By employing the completion-of-squares technique, we derive the optimal portfolio strategy and the efficient frontier, both of which are characterized by three systems of multi-dimensional ordinary differential equations (ODEs). Among these, two systems are linear, while the first one is an $\ell$-dimensional, fully coupled, and highly nonlinear Riccati equation. In the absence of regime-switching-induced stock price shocks, these systems reduce to simple linear ODEs. Thus, the introduction of regime-switching-induced stock price shocks adds significant complexity and challenges to our model. Additionally, we explore the MV problem under a no-shorting constraint. In this case, the corresponding Riccati equation becomes a $2\ell$-dimensional, fully coupled, nonlinear ODE, for which we establish solvability. The solution is then used to explicitly express the optimal portfolio and the efficient frontier.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is mathematically dense, featuring multi-dimensional nonlinear Riccati equations and ODE systems derived from advanced stochastic control theory, but it lacks empirical validation, code, or backtesting, focusing solely on theoretical model formulation and solvability.

Research Flowchart

  flowchart TD
  A["Research Goal: Optimal MV Portfolio with Regime-Switching Jumps"] --> B["Model Setup & Data Inputs"]
  B --> C["Methodology: Completion-of-Squares & ODE Systems"]
  C --> D{"Regime-Switching Shock?"}
  D -- Yes --> E["Complex Solvability: Coupled Nonlinear Riccati Equation"]
  D -- No --> F["Simpler Solvability: Linear ODEs"]
  E --> G["Key Findings: Explicit Optimal Portfolio & Efficient Frontier"]
  F --> G