Consumption and Investment with Multiple Job-Switching Opportunities

Paper: arXiv 2611.00900

Authors: Gugyum Ha, Junkee Jeon

Abstract

We study finite-horizon optimal switching with affine rewards in a consumption-investment model with finitely many occupational regimes. Directed shortest-path costs yield the exact natural borrowing limit. A dual transformation reduces the control problem to a one-dimensional system of variational inequalities with interconnected obstacles. Convexity and a distributional waiting inequality give local spatial $C^{1,1}$ regularity and smooth fit, including destination ties. These estimates establish strong duality and recover optimal consumption, investment, and switching strategies. A reflected backward stochastic differential equation representation yields a set-based integral formula without prescribed boundary geometry. Every connected component of the spatially regular locus is a maximal smooth graph over time. Exact examples exhibit disconnected contact sets, temporal boundaries under strict triangle costs, and a corner in an individual boundary component under additive route costs. A convergent time discretisation completes the analysis.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly complex mathematical framework for optimal consumption and investment with job switching, demonstrating significant theoretical novelty. However, it lacks empirical validation or backtesting, focusing purely on theoretical derivations and examples. The clarity is moderate due to the dense mathematical content, but the methodology is well-detailed for theoretical replication.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Consumption-Investment with Multiple Job-Switching] --> B{Key Methodology: Optimal Switching & Variational Inequalities};
    B --> C[Inputs: Finite Horizon, Affine Rewards, Finitely Many Occupational Regimes];
    C --> D{Computational Processes: Dual Transformation, Reflected BSDE, Time Discretization};
    D --> E[Key Findings: Exact Borrowing Limit, Spatial C1,1 Regularity, Smooth Fit, Strong Duality];
    E --> F[Outcomes: Optimal Strategies, Integral Formula, Disconnected Contact Sets, Convergent Discretization];