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Econ-95 · economics · Economics

Discounted continuous-time turnpikes and the discount-dependent reference

LLM status Open unverified

Statement

SC-11 |  Stochastic control and dynamic optimization

Research agenda formalization

Definitions and assumptions

Fix $\rho>0$. On compact state and control constraints, consider \[ \min\left\{\int_0^T e^{-\rho t}\ell(x(t),u(t))dt +e^{-\rho T}\Phi(x(T))\right\},\qquad \dot x=f(x,u),\quad x(0)=x_0. \] Assume $f,\ell,\Phi$ are continuous, $f$ is uniformly Lipschitz in $x$, and the admissible family has optimal solutions. Terminal costs are uniformly bounded in the family. A candidate reference is a discount-dependent stationary pair $(\bar x_\rho,\bar u_\rho)$. In a differentiable interior case, its current-value costate $\bar\lambda_\rho$ should satisfy \[ f(\bar x_\rho,\bar u_\rho)=0,\quad \ell_u+f_u^\top\bar\lambda_\rho=0,\quad \ell_x+f_x^\top\bar\lambda_\rho=\rho\bar\lambda_\rho, \] with derivatives evaluated at the reference pair.

Formal question

Characterize assumptions guaranteeing an appropriate reference and, for every $\varepsilon>0$, an unweighted measure-turnpike bound \[ \operatorname{Leb}\{t\in[0,T]: |x_T(t)-\bar x_\rho|+|u_T(t)-\bar u_\rho|>\varepsilon\} \leq C_\varepsilon(\rho), \] uniform over $T$ and the specified endpoint/terminal-cost family. Determine the dependence of these constants, and any exponential decay rates, on $\rho$, including conditions for uniformity as $\rho\downarrow0$.

Scope and status

The reference need not minimize the undiscounted static running cost subject to $f=0$. A discounted-weight measure bound alone is too weak: $\int_0^T e^{-\rho t}dt\leq1/\rho$ holds regardless of trajectory. The source identifies a continuous-time discounted research direction; the displayed target is a precise formalization, rather than a universal conjecture for every discounted problem.

References for the source of the problem

Faulwasser and Grüne, Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results, arXiv:2011.13670v3, §5, "Topics not Discussed and Open Problems," p. 25, first paragraph on discounting; §2 for turnpike definitions. https://arxiv.org/pdf/2011.13670v3.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-95-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-95

Discounted continuous-time turnpikes and the discount-dependent reference: An explicit discounted reference and two boundary-layer rates

Question and a solvable family

The canonical question concerns a reference depending on the discount rate and an unweighted turnpike bound: \[ \operatorname{Leb}\{t\in[0,T]:|x_T(t)-\bar x_\rho| +|u_T(t)-\bar u_\rho|>\epsilon\} \le C_\epsilon(\rho). \] The continuous-time discounted direction is identified in [fg, Section 5]. This note solves an affine-quadratic integrator family, including terminal-cost variation and uniformity as $\rho\downarrow0$. The calculation is an explicit restricted result using classical variational methods, not a general nonlinear turnpike theorem.

Fix $q,r>0$, $a,b\in\mathbb R$, and $\rho\ge0$. For $T>0$ minimize \[ J_T(x,u)=\int_0^T e^{-\rho t} \left\{\frac q2(x(t)-a)^2+\frac r2(u(t)-b)^2\right\}dt +e^{-\rho T}c x(T),\qquad \dot x=u,\quad x(0)=x_0. \tag{1} \] The families $|x_0|\le X$ and $|c|\le C$ are fixed. Compact constraints will be imposed below by placing them outside an explicit envelope. Terminal costs in (1) are then uniformly bounded, as required in the canonical problem.

Stationary reference and exact solution

Define \[ \bar x_\rho=a+\frac{\rho r b}{q},\quad\bar u_\rho=0, \quad\bar\lambda_\rho=rb, \] \[ \beta_\rho=\frac{\sqrt{\rho^2+4q/r}-\rho}{2},\qquad \gamma_\rho=\frac{\sqrt{\rho^2+4q/r}+\rho}{2}. \] Both rates are positive, $\gamma_\rho-\beta_\rho=\rho$, and $\beta_\rho\gamma_\rho=q/r$. Put $z_0=x_0-\bar x_\rho$, $v=b-c/r$, and \[ A_T=\frac{\gamma_\rho z_0-v e^{-\gamma_\rho T}} {\gamma_\rho+\beta_\rho e^{-(\beta_\rho+\gamma_\rho)T}}, \qquad B_T=\frac{v+\beta_\rho z_0e^{-\beta_\rho T}} {\gamma_\rho+\beta_\rho e^{-(\beta_\rho+\gamma_\rho)T}}. \tag{2} \]
Theorem: Among all absolutely continuous trajectories with square-integrable controls and prescribed $x_0$, (1) has the unique minimizer \[ x_T(t)=\bar x_\rho+A_Te^{-\beta_\rho t} +B_Te^{-\gamma_\rho(T-t)},\quad u_T(t)=-\beta_\rho A_Te^{-\beta_\rho t} +\gamma_\rho B_Te^{-\gamma_\rho(T-t)}. \tag{3} \] This is also the unique solution under any compact state and control constraints containing (3).
Proof: The Euler--Lagrange equation is $r\ddot x-\rho r\dot x-q(x-a)+\rho rb=0$. The free terminal condition is $r(u(T)-b)+c=0$. Subtracting $\bar x_\rho$ gives roots $-\beta_\rho,\gamma_\rho$. The initial and terminal conditions become $A_T+B_Te^{-\gamma_\rho T}=z_0$ and $-\beta_\rho A_Te^{-\beta_\rho T}+\gamma_\rho B_T=v$. Their positive determinant yields exactly (2). For any variation $h$ with $h(0)=0$, integrate the cross term in $J_T(x_T+h,u_T+\dot h)-J_T(x_T,u_T)$ by parts. The differential equation cancels its integral and the terminal condition cancels its boundary term. The remaining difference is \[ \frac12\int_0^T e^{-\rho t}\{q h(t)^2+r\dot h(t)^2\}dt\ge0, \] with equality only for $h=0$ almost everywhere, hence everywhere by continuity. This establishes existence and uniqueness directly and also proves optimality when feasible compact constraints are added. ∎
The reference satisfies the current-value equations in the question: $u=0$, $r(u-b)+\lambda=0$, and $q(x-a)=\rho\lambda$. The undiscounted static constrained minimizer instead has $x=a,u=0$. For $\rho b\ne0$ these references differ.

Unweighted measure bound and discount dependence

Put $Z_\rho=X+|\bar x_\rho|$, $V=|b|+C/r$, and \[ K_-(\rho)=(1+\beta_\rho)(Z_\rho+V/\gamma_\rho),\qquad K_+(\rho)=(1+\gamma_\rho)(V/\gamma_\rho+ \beta_\rho Z_\rho/\gamma_\rho). \]
Corollary: Uniformly over $T>0$, $|x_0|\le X$ and $|c|\le C$, \[ |x_T(t)-\bar x_\rho|+|u_T(t)| \le K_-(\rho)e^{-\beta_\rho t} +K_+(\rho)e^{-\gamma_\rho(T-t)}. \tag{4} \] Writing $\log_+(z)=\max\{0,\log z\}$ for $z>0$ and $\log_+(0)=0$, an unweighted measure-turnpike constant is \[ C_\epsilon(\rho)=\frac{\log_+(2K_-(\rho)/\epsilon)}{\beta_\rho} +\frac{\log_+(2K_+(\rho)/\epsilon)}{\gamma_\rho}. \tag{5} \] For every finite $\rho_{\max}$, these constants are uniformly bounded for $0\le\rho\le\rho_{\max}$.
Proof: The denominator in (2) is at least $\gamma_\rho$, so $|A_T|\le Z_\rho+V/\gamma_\rho$ and $|B_T|\le V/\gamma_\rho+\beta_\rho Z_\rho/\gamma_\rho$. Insert these bounds in (3) to obtain (4). If the left side exceeds $\epsilon$, at least one summand on the right exceeds $\epsilon/2$. Those two events occupy initial and terminal intervals of lengths at most the respective terms of (5); their overlap can only reduce the union's measure. On a compact discount interval $\beta_\rho$ and $\gamma_\rho$ are continuous and strictly positive, and $Z_\rho,V,K_\pm(\rho)$ are bounded. This proves uniformity, including the limit $\beta_0=\gamma_0=\sqrt{q/r}$. ∎
Proposition: Fix $\rho_{\max}<\infty$. There are compact intervals for state and control, independent of $T,x_0,c,\rho$, containing every solution (3) for $0\le\rho\le\rho_{\max}$. Thus the preceding results hold within the compact-constraint setting of the original question.
Proof: Both exponential factors in (3) are at most one. The coefficient bounds are uniform on the compact discount interval, so the states and controls have finite uniform bounds. Choose closed intervals strictly larger than these bounds. Formula (3) is feasible and the strict-convexity proof applies. On the chosen state interval, $|\Phi(x)|=|cx|$ is uniformly bounded. ∎

Limits of the result

The two rates are asymmetric: as $\rho\to\infty$, $\beta_\rho\sim q/(r\rho)$ and $\gamma_\rho\sim\rho$. Thus even this explicit example does not provide a discount-independent initial boundary-layer rate on an unbounded discount range. For the undiscounted static reference $a$, (4) shows a further concrete failure: if $b\ne0$ and $\epsilon<|\bar x_\rho-a|/2$, then, for all large $T$, a central interval of length $T-O(1)$ is farther than $\epsilon$ from $a$. This follows by making each of the two weighted exponential summands in (4) smaller than $|\bar x_\rho-a|/4$ outside fixed boundary intervals. The full question remains open here for nonlinear dynamics, active constraints, general terminal costs, and global existence or selection of a stationary discounted reference. The derivation proves a substantial solvable family, not necessary and sufficient conditions for all those cases.

References

  • [fg] T. Faulwasser and L. Grüne. Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results, arXiv:2011.13670v3, Section 5, p. 25. https://arxiv.org/pdf/2011.13670v3.

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