Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic models

Paper: arXiv 2302.08809

Authors: Filippo de Feo, Salvatore Federico, Andrzej Święch

Abstract

In this manuscript we consider a class optimal control problem for stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a $C^{1,α}$-partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising).

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly sophisticated theoretical framework using infinite-dimensional Hilbert spaces and Hamilton-Jacobi-Bellman equations to address stochastic delay differential equations. While mathematically rigorous and novel in its partial regularity proofs, it is purely theoretical and entirely lacks empirical backtesting or data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Control of Stochastic Delay Differential Equations - SDDEs] --> B[Rewrite SDDE Problem in Infinite-Dimensional Hilbert Space]
    B --> C[Data/Inputs: Path-Dependent Financial & Economic Parameters]
    C --> D[Derive Infinite-Dimensional Hamilton-Jacobi-Bellman - HJB Equation]
    D --> E[Apply Dynamic Programming & Viscosity Solution Framework]
    E --> F[Key Findings: Unique Viscosity Solution & C1,a-Partial Regularity]
    F --> G[Outcomes: Optimal Strategies for Merton Portfolio & Advertising Models]