Paper: arXiv 2411.13558

Abstract

The strong relative arbitrage problem in Stochastic Portfolio Theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative continuous solution of a Cauchy problem for a partial differential equation (PDE). However, solving this type of PDE poses analytical and numerical challenges, due to the high dimensionality and its non-unique solutions. In this paper, we discuss numerical methods to address the relative arbitrage problem and the associated PDE in a volatility-stabilized market, using time-changed Bessel bridges. We present a practical algorithm and demonstrate numerical results through an example in volatility-stabilized markets.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is dense with advanced mathematics including high-dimensional PDEs, Cauchy problems, and stochastic differential equations, but the empirical component is limited to numerical simulations without real-world data or backtesting.

Research Flowchart

  flowchart TD
  Goal["Research Goal:<br>Find nonnegative minimal solution for Cauchy PDE<br>in volatility-stabilized markets"]

  Data["Input Data:<br>Volatility-stabilized market parameters<br>Equity price dynamics<br>Time horizon T"]

  Method["Methodology:<br>Time-changed Bessel Bridges<br>Transform PDE into heat equation"]

  Process["Computational Process:<br>Solve transformed PDE numerically<br>Apply inverse time-change"]

  Outcome["Key Outcome:<br>Identification of smallest nonnegative solution<br>Characterizes highest relative return<br>Practical numerical algorithm"]

  Goal --> Data
  Data --> Method
  Method --> Process
  Process --> Outcome

Cited References (63)

  1. Minimal solutions of master equations for extended mean field games (2023) — link
  2. On first order mean field game systems with a common noise (2022)
  3. Mean field games with common noise (2016)
  4. Sharp uniform-in-time propagation of chaos (2022)
  5. Propagation of chaos: A review of models, methods and applications. II. Applications (2021)
  6. Propagation of chaos: A review of models, methods and applications. I. Models and methods (2022)
  7. Optimal arbitrage under model uncertainty (2011)
  8. Relative arbitrage in volatility-stabilized markets (2005)
and 55 more
  1. Ergodic robust maximization of asymptotic growth (2021)
  2. Mean field control and mean field game models with several populations (2018) — link
  3. A supermartingale characterization of sets of stochastic integrals and applications (2002)
  4. A probabilistic weak formulation of mean field games and applications (2015)
  5. Closed-loop convergence for mean field games with common noise (2023) — link
  6. Relative Arbitrage Opportunities in $ N $ Investors and Mean-Field Regimes (2020) — link
  7. Mean field games of controls: on the convergence of Nash equilibria (2023)
  8. Extended mean field control problem: a propagation of chaos result (2022)
  9. Topics in propagation of chaos (1991)
  10. The variational structure and time-periodic solutions for mean-field games systems (2018) — link
  11. Optimization of relative arbitrage (2015)
  12. Functional portfolio optimization in stochastic portfolio theory (2022)
  13. Cover’s universal portfolio, stochastic portfolio theory, and the num 'e raire portfolio (2019)
  14. Universal portfolios in stochastic portfolio theory (2015) — link
  15. Universal portfolios (1991)
  16. Optimal trading strategies under arbitrage (2011)
  17. On optimal arbitrage (2010)
  18. Stochastic portfolio theory (2002)
  19. N-player and mean-field games in It ^o -diffusion markets with competitive or homophilous interaction (2022)
  20. Relative Arbitrage Opportunities in an Extended Mean Field System — → 2311.02690
  21. Finding multiple solutions of odes with neural networks (2020)
  22. Options and bubbles (2007)
  23. Bessel processes, the integral of geometric Brownian motion, and Asian options (2004)
  24. Bessel processes and a functional of Brownian motion (2004)
  25. BSDEs with singular terminal condition and a control problem with constraints (2014)
  26. The Black–Scholes equation in stochastic volatility models (2010)
  27. Mean field games (2007)
  28. A general characterization of the mean field limit for stochastic differential games (2016)
  29. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle (2006)
  30. Probabilistic theory of mean field games with applications I-II (2018)
  31. Conditional strong law of large number (2005)
  32. Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations (2020)
  33. Probability measures on metric spaces (2005)
  34. Stochastic differential equations and diffusion processes (2014)
  35. Stochastic differential equations and applications (1975)
  36. Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s (1997)
  37. Convergence of probability measures (2013)
  38. Portfolio generating functions (1999)
  39. Topics in relative arbitrage, stochastic games and high-dimensional PDEs (2021)
  40. Mean field and n-agent games for optimal investment under relative performance criteria (2019)
  41. Weak existence and uniqueness for McKean–Vlasov SDEs with common noise (2021)
  42. Stochastic mckean-vlasov equations (1995)
  43. Dynamic programming for optimal control of stochastic McKean–Vlasov dynamics (2017)
  44. Weak and strong solutions of general stochastic models (2014)
  45. Robust preferences and robust portfolio choice (2009)
  46. Portfolio optimization under model uncertainty and BSDE games (2011)
  47. Risk minimizing portfolios and HJBI equations for stochastic differential games (2008)
  48. Optimal transport: old and new (2009)
  49. Bessel and volatility-stabilized processes (2009)
  50. Mean field games of controls: Finite difference approximations (2020) — link
  51. Stochastic differential equations: an introduction with applications (2013)
  52. Mean field games: numerical methods (2010)
  53. Diversity and relative arbitrage in equity markets (2005)
  54. The exit measure of a supermartingale (1972)
  55. Particle representations for a class of nonlinear SPDEs (1999)

1 of 63 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.