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arxiv: 1907.03401 · v1 · pith:NNPYMORYnew · submitted 2019-07-08 · 🧮 math.OC

On a switching control problem with c\`adl\`ag costs

Said Hamad\`ene , H\'ector Jasso-Fuentes , Yamid A. Osorio-Agudelo This is my paper

Pith reviewed 2026-05-25 01:30 UTC · model grok-4.3

classification 🧮 math.OC
keywords switching controlcàdlàg costsbackward stochastic differential equationsε-optimal policiesoptimal cost functionquasi-variational inequalitiesHJB equations
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The pith

Switching control problems with càdlàg costs admit multiple characterizations of the optimal cost function together with ε-optimal policies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies switching control problems in which the costs of changing between regimes are allowed to be discontinuous in time, specifically of càdlàg type. It establishes several equivalent descriptions of the optimal cost function and shows that control policies whose total cost lies within any prescribed epsilon of the optimum can be found. As a byproduct the work proves existence and uniqueness for a system of backward stochastic differential equations whose obstacles are càdlàg and depend on the unknown solution. When the underlying state includes a diffusion process, the optimal payoff is shown to satisfy a system of quasi-variational HJB partial differential equations in the weak sense. The results extend earlier analyses that required the costs to be continuous.

Core claim

The optimal cost function for a switching control problem with càdlàg costs can be characterized through solutions to systems of BSDEs with càdlàg obstacles that depend on the solution itself, and ε-optimal control policies exist. When an underlying diffusion is present, the optimal payoff becomes a weak solution of the associated HJB system of PDEs with obstacles of quasi-variational type.

What carries the argument

A system of backward stochastic differential equations whose barriers are càdlàg and depend on the unknown solution.

If this is right

  • The optimal cost can be obtained by solving the system of BSDEs with càdlàg obstacles.
  • ε-optimal policies can be constructed from the solution of that BSDE system.
  • Dynamic programming principles continue to hold for the value function despite jumps in the costs.
  • When a diffusion drives the state, the value function satisfies the quasi-variational HJB system in the weak sense.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same BSDE approach could be tested on impulse control problems where discontinuities arise from lump-sum payments.
  • Numerical approximation schemes for càdlàg-obstacle BSDEs might be developed by discretizing the jumps while preserving the dependence on the solution.
  • The weak-solution result for the PDE system suggests that viscosity-solution techniques could be compared directly with the probabilistic characterization.

Load-bearing premise

The cost processes are càdlàg and the filtration satisfies the standard conditions needed to apply BSDE theory and dynamic programming to switching problems.

What would settle it

An explicit switching control problem with càdlàg costs for which the associated BSDE system has no solution, or for which no ε-optimal policy exists, would falsify the central claims.

read the original abstract

This work addresses a switching control problem under which the cost associated with the changes of regimes is allowed to have discontinuities in time. Our main contribution is to show several characterizations of the optimal cost function as well as the existence of "-optimal control policies. As a by-product, we also study the existence and uniqueness of solutions of a system of backward stochastic differential equations whose barriers (or obstacles) are discontinuous (in fact of c\`adl\`ag type) and depend itself on the unknown solution. At the last part of the paper, we study the case when an underlying diffusion is part of the dynamic of the system. In this special case, the optimal payoff becomes a weak solution of the HJB system of PDEs with obstacles which is of quasi-variational type. This paper is somehow a continuation of the papers [8, 17] that consider continuous costs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript addresses a switching control problem in which regime-switching costs are allowed to be càdlàg processes. It establishes several characterizations of the optimal cost function together with the existence of ε-optimal control policies. As a by-product it proves existence and uniqueness for a system of BSDEs whose càdlàg barriers depend on the unknown solution. In the special case with an underlying diffusion the value function is shown to be a weak solution of a quasi-variational HJB system with obstacles. The work is presented as a continuation of the continuous-cost results in references [8,17].

Significance. If the derivations hold, the extension of switching-control theory to càdlàg costs is a meaningful advance for applications that involve jumps or discontinuities. The BSDE result with solution-dependent càdlàg obstacles constitutes an independent technical contribution. The diffusion-case link to quasi-variational inequalities supplies a concrete bridge to deterministic PDE theory. The manuscript appropriately positions itself as building on the cited continuous-cost papers.

minor comments (3)
  1. [Abstract] Abstract: the phrase “existence of “-optimal control policies” contains a typographical error; replace with ε-optimal.
  2. [Abstract] Abstract: the clause “whose barriers … depend itself on the unknown solution” is grammatically awkward; rephrase to “whose barriers depend on the unknown solution itself”.
  3. [Introduction] The manuscript should state explicitly in the introduction which of the characterizations in the continuous-cost papers [8,17] carry over verbatim and which require new arguments because of the càdlàg assumption.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision for our manuscript extending switching control to càdlàg costs, including the BSDE and weak HJB results. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior results via standard methods

full rationale

The paper positions itself as a continuation of [8,17] for the continuous-cost case but derives new characterizations of the optimal cost function, existence of ε-optimal policies, and BSDE solutions with càdlàg obstacles directly from the discontinuous setting using BSDE theory, dynamic programming, and (in the diffusion case) HJB quasi-variational inequalities. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central claims rest on external stochastic control techniques and the càdlàg extension rather than tautological renaming or imported uniqueness from the authors' prior work alone.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; standard assumptions of stochastic control (e.g., progressive measurability, càdlàg paths) are implicit but not detailed.

pith-pipeline@v0.9.0 · 5689 in / 930 out tokens · 33594 ms · 2026-05-25T01:30:16.006266+00:00 · methodology

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