Well-posedness and completion of the Hamilton–Jacobi theory for the Achdou–Giraud–Lasry–Lions long-term mining model

The 2016 paper of Achdou, Giraud, Lasry and Lions reduced the mean-field equilibrium of a long-term mining industry to low-dimensional Hamilton–Jacobi equations that are degenerate at the boundary $y=0$ (vanishing reserve-to-demand ratio) and whose right-hand side may blow up there. The authors proved comparison results for a first-order model problem and, for singular data, the existence of a minimal solution only in the regime $0<\alpha<m-1$ (Proposition 3.2), analyzed the demand-driven equation only in the case $\sigma=b=0$ (Theorem 3.4), stated without proof the expected blow-up behavior in the regime $\alpha\ge m-1$ (Remark 3.3), formulated but did not analyze the two-technology system, and deferred all economic interpretation to an unpublished companion paper. The problem is to complete this program: (i) rigorous well-posedness for the full degenerate second-order equation with diffusion $\sigma>0$ near the singular boundary; (ii) existence, uniqueness, and blow-up characterization for the model problem $v+H(x,v')=x^{-\alpha}$ in the regime $\alpha\ge m-1$; (iii) an existence, uniqueness, and regularity theory for the two-technology system and its scalar potential formulation; and (iv) the economic interpretation and comparative statics of the calibrated model. As of 2026, no published work closes any component of this program; the problem remains open.

Background

Mean field games (MFG), introduced by Lasry and Lions (2007), describe Nash equilibria of differential games with a continuum of identical players: each player solves a stochastic optimal-control problem whose value depends on the distribution of the other players' states, and at equilibrium the individual optimal feedbacks reproduce that distribution. When a source of randomness is common to all players (common noise), the equilibrium is governed by a master equation, a partial differential equation set on a space of probability measures, whose well-posedness is known only in special cases (Cardaliaguet, Delarue, Lasry and Lions, 2015). Low-dimensional reductions of such equilibria are therefore valuable.

Achdou, Giraud, Lasry and Lions (2016) constructed such a reduction for the long-term dynamics of a mining industry. Each production unit holds a reserve $R$ of ore and chooses two controls: a production rate and a flux invested into prospection, which creates new reserves. The market price is fixed by matching total production with a demand of size $X$, whose logarithm follows a Brownian motion with drift $b$ and volatility $\sigma$ — a shock common to all producers. Because the model is homogeneous of degree zero in $(R,X)$, the unit value takes the form $u(R,X)=V'(y)$ with the reduced variable $y=R/X$, and the equilibrium reduces to a low-dimensional equation which can be integrated into a Hamilton–Jacobi (HJ) equation for the potential $V$. For a power-law demand $\tilde D(p)=p^{-s}$ and prospection efficiency $\phi(a)=C\sqrt{a}$, this equation (equation (24) of the paper) has the structure

$$(r-b)V+byV'+H_{\mathrm{dem}}(y,V')-\frac{C^2}{4}y(V')^2-\frac{\sigma^2y^2}{2}V''=0,$$

where $H_{\mathrm{dem}}$ is an explicitly given Hamiltonian that is piecewise defined: it switches between market regimes in which the industry produces at full capacity and regimes in which production is partial and the price sits at the marginal cost of the last active unit. After the change of variables $v=-V$, $x=y$, and setting $\sigma=b=0$, the equation takes the form (equation (48) of the paper)

$$r v+H_{\mathrm{pro}}(x,v')+H_{\mathrm{dem}}(x,v')=\frac{s}{1-s}k^{1-1/s}x^{1-1/s},$$

where $r>0$ is the discount rate, $1/2<s<1$, the term $H_{\mathrm{pro}}(x,p)=\gamma xp^2$ (with $\gamma>0$) comes from the prospection control, and $H_{\mathrm{dem}}$ is the piecewise demand-matching term. Each symbol denotes a single object: $r$ is the discount rate, $\alpha$ is the singular exponent of the model problem $v+H(x,v')=x^{-\alpha}$ and of the regime condition $\alpha\ge m-1$, $a$ is the prospection intensity in $\phi(a)=C\sqrt{a}$, and $H_{\mathrm{dem}}$ and $H_{\mathrm{pro}}$ are the demand-matching and prospection Hamiltonians in both (24) and (48). (The paper writes $\alpha$ both for the discount rate of (48) and for the prospection intensity, and exchanges its labels $H_1,H_2$ between (24) and (48); the distinct names above replace those reused symbols.) Since $1-1/s<0$ for $1/2<s<1$, the right-hand side blows up at $x=0$. Two structural difficulties are concentrated at this boundary: the diffusion coefficient $\frac{\sigma^2x^2}{2}$ degenerates quadratically, and the forcing is singular.

The paper's mathematical core is a viscosity-solution analysis. A viscosity solution (in the Crandall–Lions sense; see the book of Bardi and Capuzzo-Dolcetta, 1997) is a possibly nonsmooth function that satisfies the equation in the sense that smooth test functions touching it from above yield the subsolution inequality and from below the supersolution inequality; BUC denotes the class of bounded uniformly continuous functions. Boundary conditions of state-constraint type — where the state cannot leave the domain, so no boundary data is imposed but the equation constrains the solution at the boundary — are subtle (Soner, 1986; Capuzzo-Dolcetta and Lions, 1990), and for equations with singular boundary data Lasry and Lions (1989) developed a model theory of nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. For the model problem

$$v+H\Big(x,\frac{dv}{dx}\Big)=x^{-\alpha}\quad\text{in }[0,\infty),$$

with a continuous Hamiltonian satisfying assumptions (H0)–(H4) — the key ones being one-sided growth $H(x,p)\le H(x,0)+\mu x|p|^m+C_1(x)$ with $m\ge1$ and a splitting of $H$ near $x=0$ into a uniformly continuous part and a part monotone in $x$ — the paper proves comparison (Theorem 3.1 for $m=1$, Theorem 3.3 for $m>1$), existence of a minimal BUC viscosity solution for $m>1$ and $0<\alpha<m-1$ (Proposition 3.2, via a monotone limit of solutions with truncated right-hand sides), and a nonstandard boundary condition (Definition 3.1) forbidding $C^1$ test functions touching $v$ from below at $0$. Theorem 3.4 proves existence and uniqueness of a nonincreasing BUC viscosity solution of the demand-driven equation (equivalently (48)) only in the case $\sigma=b=0$. The case $\sigma>0$ is handled by a purely formal argument that the diffusion term does not change the leading boundary singularity because $x(v')^2$ dominates $x^2v''$ near $x=0$; and the regime $\alpha\ge m-1$ of the model problem, in which the solution is expected to blow up at $0$ (like $x^{(m-1-\alpha)/m}$ if $\alpha>m-1$, logarithmically if $\alpha=m-1$), is stated in Remark 3.3 but not analyzed: the authors note that uniqueness there would require a strategy different from their comparison proofs, close to the approach of Lasry and Lions (1989).

For an industry with two technologies (two production costs $c_1<c_2$, two prospection programs), the same modeling yields a coupled system of partial differential equations for the unit values $v_1(y_1,y_2)$ and $v_2(y_1,y_2)$ (equation (55)), and, when the two discount rates coincide ($r_1=r_2$), a single degenerate second-order Hamilton–Jacobi equation for a scalar potential $V$ (equation (56)) whose gradient $DV$ is expected to recover $(v_1,v_2)$ — an instance of a potential game, in which equilibria can be found by solving a central-planner problem. The paper states a monotonicity property of the two Hamiltonians (57) and computes numerical solutions for two parameter sets (observing that $v_1$ and $v_2$ blow up at the origin and that the regime partition of the piecewise Hamiltonian has up to eight zones, seven of which coexist in the first computed case), but proves no existence, uniqueness, or regularity result for either (55) or (56). Finally, the single-technology model is calibrated to 40 years of monthly commodity prices (seven parameters, fitted by maximum likelihood through inversion of the price equation and an Euler–Maruyama — a standard simulation scheme for stochastic differential equations — discretization of the reduced reserve dynamics), while the economic interpretation, including the justification of treating resources as effectively inexhaustible and of the stationarity-up-to-scale assumption, is deferred to a companion paper cited only as work in progress, which never appeared.

Problem Statement

Complete the analytical and economic theory of the Hamilton–Jacobi equations arising in the long-term mean-field model of mining industries of Achdou, Giraud, Lasry and Lions (2016). The four components below are scientifically linked: they concern the same model, the same degenerate boundary mechanism at $y=0$, and the same goal of turning a partially analyzed equilibrium model into a well-posed and economically interpretable object.

(1) Well-posedness with diffusion. For the closed single-technology industry with power-law demand (equation (24) of the paper, in the potential form displayed in the Background; its first-order specialization with $\sigma=b=0$ is equation (48) after the change of variables $v=-V$, $x=y$), prove existence and uniqueness of the viscosity solution when $\sigma>0$, for the full degenerate second-order equation (24), with the drift term retained (i.e. without the specialization $\sigma=b=0$ covered by Theorem 3.4) and containing the term $\frac{\sigma^2x^2}{2}V''$, together with the correct boundary condition at the singular boundary $x=0$ — or, alternatively, establish a concrete failure of well-posedness of (24) for $\sigma>0$, for instance non-uniqueness of the viscosity solution. Theorem 3.4 of the paper covers only $\sigma=b=0$, and the paper's argument that the diffusion term does not change the leading boundary singularity is explicitly formal.

(2) The regime $\alpha\ge m-1$. For the model problem $v+H(x,v')=x^{-\alpha}$ on $[0,\infty)$ under assumptions (H0)–(H4) with $m>1$, prove or disprove that for $\alpha\ge m-1$ there exists a unique viscosity solution in $C(0,\infty)$ blowing up at $0$ — like $x^{(m-1-\alpha)/m}$ if $\alpha>m-1$ and logarithmically if $\alpha=m-1$ — and characterize the boundary singularity precisely, including the behavior of the minimal solution obtained as the monotone limit of the truncated problems used in the paper. Remark 3.3 of the paper asserts this outcome without proof and states that uniqueness requires a strategy different from the paper's comparison proofs, close to the Lasry–Lions (1989) approach to singular boundary conditions.

(3) Two-technology theory. For the two-technology closed-industry system (equation (55)) and, when $r_1=r_2$, the scalar degenerate second-order Hamilton–Jacobi equation for the potential $V$ (equation (56)): prove existence, uniqueness, and regularity of solutions under explicit assumptions; establish or refute the potential-game reduction, i.e. the existence of a solution $V$ of (56) with nonnegative partial derivatives whose gradient $DV$ solves (55); and determine what role the monotonicity property (57) of the Hamiltonians plays in well-posedness. At present only numerical solutions for two parameter sets exist, with no analytical results.

(4) Economic interpretation. Provide the deferred qualitative and quantitative economic analysis of the calibrated model: interpretation of the fitted parameters, comparative statics with respect to production costs, capacity, prospection efficiency, and demand elasticity, and a justification of the structural approximations — resources treated as effectively inexhaustible on the modeling horizon, and stationarity up to a scale factor — which the source paper defers to a companion work that was never published. Determine, from the calibrated equilibrium, the sign of the comparative statics of the equilibrium price and value with respect to each of the four parameters — production cost, capacity, prospection efficiency, and demand elasticity — together with the limiting behavior where the model determines it, and, for each structural approximation, state the parameter regime or horizon under which it is valid and the error it introduces.

References

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