Paper: arXiv 2410.13103

Authors: Alberto Gennaro, Thibaut Mastrolia

Abstract

We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. This uncertainty introduces significant complexities in analyzing the problem, requiring distinct mathematical approaches for two cases: when the random default time falls within the initial time frame [0,T] and when it extends beyond this period. We develop a theoretical framework to model the stochastic dynamics of the investment process, incorporating the random default time. We then analyze the portfolio manager’s investment decisions and compensation mechanisms for both scenarios. In the first case, where the default time could be unbounded, we apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem. In the second case, where the default time is within the interval [0,T], the problem becomes more intricate due to the degeneracy of the BSDE’s driver. For both scenarios, we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equations in both situations. We develop a deep-learning algorithm to solve the problem in high-dimension with no access to the optimizer of the Hamiltonian function.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical formulation of the continuous-time Principal-Agent problem under random default horizons using BSDEs, HJB equations, and filtration enlargement. However, it lacks empirical validation on real-world market data, relying instead on high-dimensional PINN deep-learning algorithms for numerical solutions.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Portfolio Delegation under Random Default Time] --> B[Model Setup & Scenario Split]
    
    B -->|Scenario 1: Default unbounded| C[BSDEs & Stochastic Control Theory]
    B -->|Scenario 2: Default in 0,T| D[Integro-Partial HJB Equations]
    
    C --> E[Deep Learning Algorithm for High-Dimensional HJB]
    D --> E
    
    E --> F[Outcomes: Contract Resolution & Numerical Solutions without Hamiltonian Optimizer]

Cited References (65)

  1. Solutions of BSDE’s with jumps and quadratic/locally Lipschitz generator (2016)
  2. Enlargement of filtration with finance in view (2017)
  3. Wellposedness of second order backward SDEs (2012)
  4. Second order BSDEs with jumps by measurable selection argument (2024)
  5. Moral hazard under ambiguity (2018)
  6. Uniqueness of viscosity solutions for a class of integro-differential equations (2015)
  7. Viscosity solutions of nonlinear integro-differential equations (1996)
  8. Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions (2009)
and 57 more
  1. On pathwise stochastic integration (1995)
  2. Optimal risk-sharing with effort and project choice (2007)
  3. Random horizon principal-agent problems (2022)
  4. Second order backward SDE with random terminal time (2020)
  5. Mind the jumps: when 2BSDEs meet semi-martingales (2025) — link
  6. Agency Problems and Adversarial Bilevel Optimization under Uncertainty and Cyber Threats (2025) — link
  7. Stochastic control for a class of nonlinear kernels and applications (2018)
  8. Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs (2007)
  9. 2BSDE with uncertain horizon and application to stochastic control in erratic environments (2025) — link
  10. Optimal investment and consumption with transaction costs (1994)
  11. Optimum consumption and portfolio rules in a continuous-time model (1975)
  12. Viscosity solutions of Hamilton-Jacobi equations (1983)
  13. Continuous-time principal-agent problem with drift and stochastic volatility control: with applications to delegated portfolio management (2014)
  14. A note on BSDEs with singular driver coefficients (2014)
  15. User’s guide to viscosity solutions of second order partial differential equations (1992)
  16. Controlled Markov processes and viscosity solutions (2006)
  17. Optimal stochastic control, stochastic target problems, and backward SDE (2012)
  18. The flash crash: High-frequency trading in an electronic market (2017)
  19. On “black swans” and “perfect storms”: Risk analysis and management when statistics are not enough (2012)
  20. Existence and uniqueness results for BSDE with jumps: the whole nine yards (2018)
  21. Credit risk: modeling, valuation and hedging (2013)
  22. Mathematical methods for financial markets (2009)
  23. Backward stochastic differential equations and quasilinear parabolic partial differential equations (2005)
  24. Backward stochastic differential equations and integral-partial differential equations (1997)
  25. Adapted solution of a backward stochastic differential equation (1990)
  26. Dynamic programming approach to principal–agent problems (2018)
  27. Incentive contracts in delegated portfolio management (2009)
  28. Optimal security design and dynamic capital structure in a continuous-time agency model (2006)
  29. Delegated portfolio management: A survey of the theoretical literature (2006)
  30. Multiagent Relative Investment Games in a Jump Diffusion Market with Deep Reinforcement Learning Algorithm (2025)
  31. Optimal contracts in a continuous-time delegated portfolio management problem (2003)
  32. The agency theory applied to the investment funds (2004)
  33. The performance of mutual funds in the period 1945-1964 (1968)
  34. Investment management (1998)
  35. Optimal execution of portfolio transactions (2001)
  36. Governmental incentives for green bonds investment (2022)
  37. A new approach to principal-agent problems with volatility control (2024) — link
  38. Market making and incentives design in the presence of a dark pool: a deep reinforcement learning approach (2019) — link
  39. Optimal investment decisions when time-horizon is uncertain (2008)
  40. Changes of filtrations and of probability measures (1978)
  41. Moral hazard in dynamic risk management (2017)
  42. What happens after a default: the conditional density approach (2010)
  43. Intensity process and compensator: A new filtration expansion approach and the Jeulin–Yor theorem (2008)
  44. Aggregation and linearity in the provision of intertemporal incentives (1987)
  45. Utility maximization in incomplete markets (2005)
  46. Stochastic differential equations and diffusion processes (2014)
  47. Utility maximization with random horizon: a BSDE approach (2015)
  48. Mean-variance hedging on uncertain time horizon in a market with a jump (2013)
  49. Backward stochastic differential equations and partial differential equations with quadratic growth (2000)
  50. Utility maximization in a jump market model (2009)
  51. Backward stochastic differential equations in finance (1997)
  52. Introduction to stochastic finance with market examples (2022)
  53. Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations (2017) — link
  54. A continuous-time version of the principal-agent problem (2008)
  55. DGM: A deep learning algorithm for solving partial differential equations (2018)
  56. Foundations of portfolio theory (1991)
  57. Automatic differentiation in machine learning: a survey (2018)

0 of 65 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.