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On the Singular Control of a Diffusion and Its Running Infimum or Supremum

Giorgio Ferrari, Neofytos Rodosthenous

TL;DR

The paper develops a rigorous framework for singular stochastic control of a diffusion together with its running infimum (and, by duality, running supremum), introducing two novel integral operators that are consistent with the HJB equation in a two-dimensional state space. A general verification theorem is established for problems with state-dependent running costs, control costs, and running infimum (or supremum) costs, enabling identification of optimal controls via smooth solutions to a variational inequality with gradient constraints. The theory is applied to an optimal dividend problem where the manager’s time-preference depends multiplicatively on the company’s historical worst performance; the solution features a running-infimum dependent free boundary and a distinctive control policy that combines lump-sum dividends with reflection along a curved boundary. Additionally, the paper extends the integral framework to the running supremum setting, preserving consistency with the HJB equation. The results enrich the singular control literature by handling complex state interactions and providing explicit solutions and policy structures in economically meaningful models, with convergence to De Finetti’s classical results as the performance sensitivity vanishes.

Abstract

We study a class of singular stochastic control problems for a one-dimensional diffusion $X$ in which the performance criterion to be optimised depends explicitly on the running infimum $I$ (or supremum $S$) of the controlled process. We introduce two novel integral operators that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional singular control problems. The first operator involves integrals where the integrator is the control process of the two-dimensional process $(X,I)$ or $(X,S)$; the second operator concerns integrals where the integrator is the running infimum or supremum process itself. Using these definitions, we prove a general verification theorem for problems involving two-dimensional state-dependent running costs, costs of controlling the process, costs of increasing the running infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an optimal dividend problem in which the manager's time-preferences depend on the company's historical worst performance.

On the Singular Control of a Diffusion and Its Running Infimum or Supremum

TL;DR

The paper develops a rigorous framework for singular stochastic control of a diffusion together with its running infimum (and, by duality, running supremum), introducing two novel integral operators that are consistent with the HJB equation in a two-dimensional state space. A general verification theorem is established for problems with state-dependent running costs, control costs, and running infimum (or supremum) costs, enabling identification of optimal controls via smooth solutions to a variational inequality with gradient constraints. The theory is applied to an optimal dividend problem where the manager’s time-preference depends multiplicatively on the company’s historical worst performance; the solution features a running-infimum dependent free boundary and a distinctive control policy that combines lump-sum dividends with reflection along a curved boundary. Additionally, the paper extends the integral framework to the running supremum setting, preserving consistency with the HJB equation. The results enrich the singular control literature by handling complex state interactions and providing explicit solutions and policy structures in economically meaningful models, with convergence to De Finetti’s classical results as the performance sensitivity vanishes.

Abstract

We study a class of singular stochastic control problems for a one-dimensional diffusion in which the performance criterion to be optimised depends explicitly on the running infimum (or supremum ) of the controlled process. We introduce two novel integral operators that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional singular control problems. The first operator involves integrals where the integrator is the control process of the two-dimensional process or ; the second operator concerns integrals where the integrator is the running infimum or supremum process itself. Using these definitions, we prove a general verification theorem for problems involving two-dimensional state-dependent running costs, costs of controlling the process, costs of increasing the running infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an optimal dividend problem in which the manager's time-preferences depend on the company's historical worst performance.

Paper Structure

This paper contains 15 sections, 9 theorems, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup for a state-space involving the running infimum process
  3. The controlled process $(X^\mathrm{D}, I^\mathrm{D})$
  4. The Optimal Control Problem
  5. Integral with respect to $\mathrm{D}$
  6. Integral with respect to $I^\mathrm{D}$
  7. Stochastic control problem formulation
  8. The Hamilton-Jacobi-Bellman Equation and the Verification Theorem
  9. A problem of Optimal Dividend Payments with Multiplicative Time-Preference Impact
  10. Special case: No multiplicative time-preference impact
  11. Dividend problem with multiplicative time-preference impact: Case of $\mu\leq 0$.
  12. Dividend problem with multiplicative time-preference impact: Case of $\mu > 0$.
  13. Construction of the candidate value function
  14. Construction of the candidate optimal control
  15. Integrals for state-spaces involving the running supremum process

Key Result

Lemma 3.4

For any $(x,i) \in \mathcal{M}$ and $\mathrm{D} \in \mathcal{A}(x,i)$, we note that:

Figures (2)

  • Figure 1: Schematic depiction of the state-space $\mathcal{M}^0 = \mathcal{D}$ given by \ref{['eq:setMa-div']} and identifying with the action region $\mathcal{D}$ for the problem \ref{['eq:value-div']} with $\mu \leq 0$, and the movement (blue) of the process $(X^{x,\rm D},I^{x,\rm D})$ started from $(x,i) \in \mathcal{D}$ such that $x>i$, which jumps downwards in a 'hockey-stick' direction to the origin, where it is absorbed.
  • Figure 2: Schematic depiction of the movement (blue) of the process $(X^{x,\rm D},I^{x,\rm D})$ started from $(x,i) \in \mathcal{M}^0 = \mathcal{C} \cup \mathcal{D}_1 \cup \mathcal{D}_2$ given by \ref{['eq:setMa-div']}, \ref{['CD']} and \ref{['eq:stopping']} for the problem \ref{['eq:value-div']} with $\mu>0$. Top left panel: The process $(X^{x,\rm D},I^{x,\rm D})$ started from $(x,i)\in \mathcal{C}$ is reflected either downwards at the boundary function $x = b(i)$ (red) or in the south-west direction at the diagonal $x=i$ (black), until it is absorbed at the origin. Top right panel: The process $(X^{x,\rm D},I^{x,\rm D})$ started from $(x,i)\in \mathcal{D}_1$ jumps downwards to $(b(i),i)$ at time $0$ and then continues as in top left panel. Bottom left panel: The process $(X^{x,\rm D},I^{x,\rm D})$ started from $(x,i)\in \mathcal{D}_2$ jumps downwards in the south-west direction to $(b(i^\star),i^\star)$ at time $0$ and then continues as in top left panel. Bottom right panel: The process $(X^{x,\rm D},I^{x,\rm D})$ started from $(x,i)\in \mathcal{D}_2$ jumps downwards in a 'hockey-stick' direction to $(b(i^\star),i^\star)$ at time $0$ and then continues as in top left panel.

Theorems & Definitions (24)

  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Theorem 4.1: Verification Theorem
  • proof
  • ...and 14 more