Paper: arXiv 2610.00998

Authors: Tae Ung Gang, Donghan Kim

Abstract

We study finite-horizon portfolio optimization with proportional transaction costs and trading opportunities arriving at the jump times of a Cox process. Borrowing and short-selling are prohibited, while utility functions need not be concave, increasing, or differentiable. The admissible class includes differentiable utilities with asymptotic elasticity greater than or equal to one. The associated Hamilton–Jacobi–Bellman equation is semi-linear and nonlocal, with control entering only through a zeroth-order term. We show that the normalized equation admits a unique bounded solution that is continuous in all state variables and classical in time and log-price. The proof combines a contraction argument with interior Schauder estimates, and verification identifies the value function and yields an optimal Markovian feedback strategy. Numerical examples with non-concave utilities exhibit, at fixed time and pre-trade wealth, multiple intervals associated with the same trading action, departures from the usual buy–no-trade–sell ordering, and abrupt switches in the optimal post-trade risky weight.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly complex mathematical framework for portfolio optimization under non-standard conditions. While the theoretical contributions are significant and novel, the empirical rigor is limited to numerical examples rather than extensive data validation or backtesting. The overall score reflects the strong theoretical foundation and novelty, balanced by the lack of real-world empirical validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Portfolio Optimization with Transaction Costs and Search Frictions] --> B{Methodology: HJB Equation, Contraction Argument, Schauder Estimates};
    B --> C[Inputs: General Utility Functions, Proportional Transaction Costs, Cox Process Trading Opportunities];
    C --> D{Computational Process: Solving Semi-Linear Nonlocal HJB Equation};
    D --> E[Outcomes: Unique Bounded Continuous Solution, Optimal Markovian Feedback Strategy];
    E --> F[Key Findings: Multiple Trading Action Intervals, Departures from Buy-No-Trade-Sell, Abrupt Switches in Optimal Risky Weight];

Cited References (42)

  1. Nonconcave Utility Maximization with Transaction Costs (2026) — link
  2. Nonconcave Utility Maximization with Portfolio Bounds (2022) — link
  3. Utility Maximization under Trading Constraints with Discontinuous Utility (2019) — link
  4. Aspirational Utility and Investment Behavior (2025) — link
  5. Aspiration Level, Probability of Success and Failure, and Expected Utility (2008)
  6. Reaching Goals by a Deadline: Digital Options and Continuous-Time Active Portfolio Management (1999)
  7. The Utility Analysis of Choices Involving Risk (1948)
  8. The Utility of Wealth (1952)
and 34 more
  1. Between Hope and Fear: The Psychology of Risk (1987) — link
  2. Behavioral Portfolio Theory (2000)
  3. A Continuous Time Framework for Sequential Goal-Based Wealth Management (2024) — link
  4. Non-concave Expected Utility Optimization with Uncertain Time Horizon (2023) — link
  5. Prospect Theory: An Analysis of Decision under Risk (1979) — link
  6. Behavioral Portfolio Selection in Continuous Time (2008)
  7. Does Option Compensation Increase Managerial Risk Appetite? (2000)
  8. Portfolio Selection with Transactions Costs (1976)
  9. Portfolio Selection with Transaction Costs (1990)
  10. Optimal Investment and Consumption with Transaction Costs (1994)
  11. Financial Modelling with Jump Processes (2004)
  12. Partial Differential Equations of Parabolic Type (1964)
  13. Lectures on Elliptic and Parabolic Equations in H"older Spaces (1996) — link
  14. Unified Asymptotics for Investment Under Illiquidity: Transaction Costs and Search Frictions (2026) — link
  15. The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets (1999) — link
  16. Lifetime portfolio selection under uncertainty: The continuous-time case (1969)
  17. Optimum consumption and portfolio rules in a continuous-time model (1971)
  18. Optimal portfolio selection with transaction costs and finite horizons (2002)
  19. Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem (2009)
  20. Asymptotic analysis for optimal investment and consumption with transaction costs (2004)
  21. Transaction costs, trading volume, and the liquidity premium (2014)
  22. The relaxed investor and parameter uncertainty (2001)
  23. A simple model of liquidity effects (2002)
  24. Optimal portfolio of low liquid assets with a log-utility function (2006)
  25. A model of optimal consumption under liquidity risk with random trading times (2008)
  26. Optimal consumption policies in illiquid markets (2011)
  27. Optimal investment on finite horizon with random discrete order flow in illiquid markets (2011)
  28. Investment/consumption problem in illiquid markets with regime-switching (2014)
  29. Asset allocation and liquidity breakdowns: What if your broker does not answer the phone? (2010) — link
  30. Portfolio choice with illiquid assets (2014) — link
  31. Optimal investment in an illiquid market with search frictions and transaction costs (2023) — link
  32. Optimal portfolio choice under loss aversion (2004)
  33. Utility maximization with a given pricing measure when the utility is not necessarily concave (2013)
  34. A Coupled System of Integrodifferential Equations Arising in Liquidity Risk Model (2009)

0 of 42 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.