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A class of singular control problems with tipping points

Jean-Paul Décamps, Fabien Gensbittel, Thomas Mariotti, Stéphane Villeneuve

TL;DR

The paper develops a rigorous framework for a new class of tipping-point problems by embedding a random tipping threshold $Y$ into a singular stochastic control setting. By showing that the tipping time $ au_Y$ induces a bi-dimensional Markovian state $(X^L,M^L)$, the authors derive a two-dimensional HJB equation with novel nonlocal boundary conditions and prove a verification theorem. They then solve an explicit resource-extraction problem with tipping by constructing a free boundary $b(m)$ via an auxiliary one-dimensional problem and a detailed candidate value function $W$, identifying when an optimal policy exists or when $oldsymbol{ extit{epsilon}-optimal}$ strategies are necessary. The results extend stochastic control to systems with unobservable tipping points and have wide relevance for resource management under uncertainty, including environmental and financial contexts, with a complete mathematical treatment of existence, regularity, and optimality via Itô calculus on sets with corners.

Abstract

Tipping points define situations where a system experiences sudden and irreversible changes and are generally associated with a random level of the system below which the changes materialize. In this paper, we study a singular stochastic control problem in which the performance criterion depends on the hitting time of a random level that is not a stopping time for the reference filtration. We establish a connection between the value of the problem and the value of a singular control problem involving a diffusion and its running minimum. We prove a verification theorem and apply our results to explicitly solve a resource extraction problem where the random evolution of the resource changes when it crosses a tipping point.

A class of singular control problems with tipping points

TL;DR

The paper develops a rigorous framework for a new class of tipping-point problems by embedding a random tipping threshold into a singular stochastic control setting. By showing that the tipping time induces a bi-dimensional Markovian state , the authors derive a two-dimensional HJB equation with novel nonlocal boundary conditions and prove a verification theorem. They then solve an explicit resource-extraction problem with tipping by constructing a free boundary via an auxiliary one-dimensional problem and a detailed candidate value function , identifying when an optimal policy exists or when strategies are necessary. The results extend stochastic control to systems with unobservable tipping points and have wide relevance for resource management under uncertainty, including environmental and financial contexts, with a complete mathematical treatment of existence, regularity, and optimality via Itô calculus on sets with corners.

Abstract

Tipping points define situations where a system experiences sudden and irreversible changes and are generally associated with a random level of the system below which the changes materialize. In this paper, we study a singular stochastic control problem in which the performance criterion depends on the hitting time of a random level that is not a stopping time for the reference filtration. We establish a connection between the value of the problem and the value of a singular control problem involving a diffusion and its running minimum. We prove a verification theorem and apply our results to explicitly solve a resource extraction problem where the random evolution of the resource changes when it crosses a tipping point.
Paper Structure (14 sections, 14 theorems, 165 equations)

This paper contains 14 sections, 14 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. The problem formulation
  3. Markovian formulation and HJB equation
  4. A Markovian formulation
  5. Hamilton-Jacobi-Bellman equation and the verification lemma
  6. A problem of optimal resource extraction with a tipping point
  7. The auxiliary problem (\ref{['aux']})
  8. A free boundary problem
  9. Solution to the DM problem
  10. Candidate value function
  11. Value function, optimal strategy and $\epsilon$-optimal strategy
  12. Appendix
  13. A class of smooth functions on ${\mathcal{J}}$
  14. Proofs

Key Result

Proposition 1

Let $V$ be the value function of the singular control problem defined as with where $M^{L,c}$ is the continuous part of $M$. Then, we have

Theorems & Definitions (29)

  • Proposition 1
  • proof : Proof of Proposition \ref{['markovprop']}
  • Lemma 1
  • proof : Proof of Lemma \ref{['verif']}
  • Lemma 2
  • proof
  • Example 1
  • Proposition 2
  • Proposition 3
  • proof : Proof of Proposition \ref{['lemedo']}
  • ...and 19 more