An Explicit Solution for the Problem of Optimal Investment with Random Endowment

ArXiv ID: 2506.20506 “View on arXiv”

Authors: Michael Donisch, Christoph Knochenhauer

Abstract

We consider the problem of optimal investment with random endowment in a Black–Scholes market for an agent with constant relative risk aversion. Using duality arguments, we derive an explicit expression for the optimal trading strategy, which can be decomposed into the optimal strategy in the absence of a random endowment and an additive shift term whose magnitude depends linearly on the endowment-to-wealth ratio and exponentially on time to maturity.

Keywords: duality arguments, optimal investment, random endowment, Black-Scholes market, constant relative risk aversion

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper presents a highly theoretical, duality-based derivation of an explicit solution for optimal investment with random endowment, featuring dense stochastic calculus and partial differential equation theory. However, it lacks any empirical backtesting, data, or implementation details, relying purely on abstract mathematical analysis without practical performance metrics.

  flowchart TD
    A["Research Goal: Find Optimal Investment Strategy with Random Endowment"] --> B["Key Methodology: Duality Arguments"]
    B --> C["Model Setup: Black-Scholes Market & CRRA Utility"]
    C --> D["Define Value Function & Duality Problem"]
    D --> E["Derive Candidate Solution via KKT Conditions"]
    E --> F["Verify Optimality & Unique Boundary Conditions"]
    F --> G["Computational Process: Solve for Dual & Primal Processes"]
    G --> H["Outcome: Explicit Trading Strategy"]
    H --> I["Strategy Decomposition<br/>Optimal Base Strategy + Linear/Exponential Shift Term"]