Stochastic optimal self-path-dependent control: A new type of variational inequality and its viscosity solution

Paper: arXiv 2412.11383

Authors: Mingxin Guo, Zuo Quan Xu

Abstract

In this paper, we explore a new class of stochastic control problems characterized by specific control constraints. Specifically, the admissible controls are subject to the ratcheting constraint, meaning they must be non-decreasing over time and are thus self-path-dependent. This type of problems is common in various practical applications, such as optimal consumption problems in financial engineering and optimal dividend payout problems in actuarial science. Traditional stochastic control theory does not readily apply to these problems due to their unique self-path-dependent control feature. To tackle this challenge, we introduce a new class of Hamilton-Jacobi-Bellman (HJB) equations, which are variational inequalities concerning the derivative of a new spatial argument that represents the historical maximum control value. Under the standard Lipschitz continuity condition, we demonstrate that the value functions for these self-path-dependent control problems are the unique solutions to their corresponding HJB equations in the viscosity sense.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This pure mathematical finance paper establishes a new theoretical framework for stochastic optimal control under ratcheting constraints via viscosity solutions of a new class of HJB variational inequalities. While highly rigorous mathematically with complete theoretical derivations, it contains no empirical data, implementation code, or backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Solve stochastic control under<br>ratcheting constraints (non-decreasing control)"] --> B["Inputs: Financial/Actuarial Models<br>& Lipschitz Continuity Conditions"]
    B --> C["Formulate Value Function with Historical<br>Maximum Control as Spatial Argument"]
    C --> D["Derive New HJB Variational Inequality<br>concerning Spatial Argument Derivative"]
    D --> E["Prove Existence & Uniqueness of<br>Viscosity Solutions for HJB Equation"]
    E --> F["Outcomes: Rigorous Framework for<br>Stochastic Self-Path-Dependent Control"]