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arxiv: 2502.02352 · v3 · pith:X2CZNPFPnew · submitted 2025-02-04 · 🧮 math.OC · econ.GN· math.PR· q-fin.EC

Stochastic Optimal Control with Measurable Coefficients and Applications

Filippo de Feo This is my paper

Pith reviewed 2026-05-23 04:17 UTC · model grok-4.3

classification 🧮 math.OC econ.GNmath.PRq-fin.EC
keywords stochastic optimal controlmeasurable coefficientsHamilton-Jacobi-Bellman equationL^p-viscosity solutionsverification theoremsoptimal feedback controlsinfinite horizonoptimal advertising
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The pith

For stochastic optimal control with measurable coefficients, the HJB equation admits an L^p-viscosity solution in W_loc^{2,p} that satisfies the equation almost everywhere.

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

This paper aims to establish that fully nonlinear stochastic optimal control problems in infinite horizon with merely measurable coefficients still possess L^p-viscosity solutions in the local Sobolev space W_loc^{2,p} to the Hamilton-Jacobi-Bellman equation, provided the diffusion is locally uniformly elliptic. A sympathetic reader would care because these solutions turn out to be strong, satisfying the equation pointwise almost everywhere, which opens the way to verification theorems that certify optimality. The results further allow construction of optimal feedback controls and identification of the value function as the unique such solution. An application to an optimal advertising problem in economics illustrates the reach of the theory beyond smooth-coefficient settings.

Core claim

Using the theory of L^p-viscosity solutions, existence is shown of an L^p-viscosity solution v in W_loc^{2,p} of the HJB equation for fully non-linear stochastic optimal control problems with measurable coefficients and locally uniformly elliptic diffusion; this v is also a strong solution satisfying the equation a.e. Verification theorems are proved that give necessary and sufficient conditions for optimality, leading to optimal feedback controls and characterization of the value function as the unique such solution. The results are used to solve an optimal advertising problem from economics.

What carries the argument

L^p-viscosity solutions to the Hamilton-Jacobi-Bellman equation

If this is right

  • Verification theorems supply necessary and sufficient conditions for optimality.
  • Optimal feedback controls can be constructed explicitly from the solution.
  • The value function is characterized as the unique L^p-viscosity solution of the HJB equation.
  • The framework solves a stochastic optimal advertising problem arising in economics.

Where Pith is reading between the lines

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

  • The same existence result may support optimal control models in economics or finance that incorporate measured, non-smooth data.
  • Numerical schemes could be developed that approximate these L^p-viscosity solutions directly.
  • The approach suggests a route to treat other control problems whose coefficients arise from empirical estimation rather than analytic assumptions.

Load-bearing premise

The diffusion coefficient must be locally uniformly elliptic.

What would settle it

A concrete stochastic control problem with measurable coefficients and non-elliptic diffusion for which no L^p-viscosity solution exists in W_loc^{2,p}.

read the original abstract

Stochastic optimal control control problems with merely measurable coefficients are not well understood. In this manuscript, we consider fully non-linear stochastic optimal control problems in infinite horizon with measurable coefficients and (local) uniformly elliptic diffusion. Using the theory of $L^p$-viscosity solutions, we show existence of an $L^p$-viscosity solution $v\in W_{\rm loc}^{2,p}$ of the Hamilton-Jacobi-Bellman (HJB) equation, which, in turn, is also a strong solution (i.e. it satisfies the HJB equation pointwise a.e.). We are then led to prove verification theorems, providing necessary and sufficient conditions for optimality. These results allow us to construct optimal feedback controls and to characterize the value function as the unique $L^p$-viscosity solution of the HJB equation. To the best of our knowledge, these are the first results for fully non-linear stochastic optimal control problems with measurable coefficients. We use the theory developed to solve a stochastic optimal control problem arising in economics within the context of optimal advertising.

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 considers infinite-horizon fully nonlinear stochastic optimal control problems with measurable coefficients and locally uniformly elliptic diffusion. It applies the theory of L^p-viscosity solutions to establish existence of a function v in W_loc^{2,p} that solves the HJB equation in the L^p-viscosity sense and also satisfies the equation pointwise almost everywhere as a strong solution. Verification theorems are proved, optimal feedback controls are constructed from the solution, and the value function is characterized as the unique L^p-viscosity solution. The results are illustrated with an application to an optimal advertising problem arising in economics.

Significance. If the derivations hold, the work is significant as it extends stochastic control theory to a broad class of problems with discontinuous coefficients, where classical continuity assumptions fail. The approach correctly leverages the existing L^p-viscosity framework (designed precisely for fully nonlinear elliptic equations with measurable data under local uniform ellipticity) together with standard dynamic-programming arguments, without introducing new parameters or ad-hoc entities. The verification theorems and economic application add concrete value, and the results are falsifiable via the constructed feedback controls.

minor comments (3)
  1. [Introduction] Introduction: the literature comparison to prior viscosity results for HJB equations (even in deterministic or linear cases with measurable coefficients) is brief; expanding it would better highlight the precise novelty of the stochastic fully nonlinear setting.
  2. [Main existence result] The statement that v is both an L^p-viscosity solution and a strong solution a.e. would benefit from an explicit cross-reference to the precise integrability and ellipticity conditions used to pass from one notion to the other.
  3. [Application] In the optimal advertising application, the explicit measurable coefficient chosen for the example should be displayed alongside the resulting HJB equation to make the necessity of the L^p theory immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive assessment of its significance, and the recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies the pre-existing theory of L^p-viscosity solutions (under local uniform ellipticity) to obtain existence of a W_loc^{2,p} solution to the HJB equation for the infinite-horizon control problem, then invokes standard dynamic-programming arguments for verification and uniqueness. No step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the central existence claim is an application of an independent external theory rather than an internal renaming or tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of L^p-viscosity solution theory to HJB equations with measurable coefficients under local uniform ellipticity; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The theory of L^p-viscosity solutions applies to fully nonlinear HJB equations with measurable coefficients when the diffusion is locally uniformly elliptic.
    This is the key background theory invoked to prove existence of the solution.

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