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

TL;DR

This paper introduces a new class of stochastic control problems with ratcheting constraints, leading to novel variational inequality HJB equations whose viscosity solutions characterize the optimal control strategies.

Contribution

It develops a new theoretical framework for self-path-dependent control problems with ratcheting constraints, including the formulation of associated variational inequality HJB equations.

Findings

Proves the existence and uniqueness of viscosity solutions for the new HJB equations.

Establishes the value functions as the unique solutions under Lipschitz conditions.

Provides a foundation for solving practical control problems with non-decreasing control constraints.

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.