Discounted drift control with discretionary stopping

SC-08 |  Stochastic control and dynamic optimization

Research agenda formalization

Definitions and assumptions

Use the control/state and cost conventions preceding SC-07, and fix $\rho>0$. The natural discounted extension allows $\tau\in[0,\infty]$ and defines \[ V_\rho(x)=\inf_{u,\tau} \E\left[\int_0^\tau e^{-\rho s}(\psi(u_s)+c)\,ds +\ind_{\{\tau<\infty\}}e^{-\rho\tau}k(X_\tau)\right]. \] Thus the terminal cost on never stopping is, by definition, zero. The control $u=0$, with never stopping, shows $0\leq V_\rho(x)\leq\min\{k(x),c/\rho\}$ whenever its weak state system is admissible. Define $\mathcal S_\rho=\{x:V_\rho(x)=k(x)\}$.

Formal question

Characterize $V_\rho$ and optimal control/stopping rules for this discounted problem. Determine when $\mathcal S_\rho$ is an interval $[-b_\rho,b_\rho]$, and derive conditions that uniquely determine $b_\rho$ and the admissible continuation solution of \[ \min\left\{k-V_\rho,\ \tfrac12\sigma^2V_\rho''+ \inf_{a\in\R}\{a\mu V_\rho'+\psi(a)\}+c-\rho V_\rho\right\}=0. \] The characterization should specify the regularity used for smooth fit and the transversality condition needed to verify a feedback/stopping pair, including paths on which stopping never occurs.

Scope and status

Discounting changes admissibility at infinite maturity: the positive operating cost no longer forces $\E\tau<\infty$. Restricting to finite stopping times is a distinct variant that must be stated separately. The source's proposed extension is a research direction; this entry does not claim that general discounted control-and-stopping existence theory is itself wholly unresolved.

References for the source of the problem

Beneš, Gaitsgori and Karatzas, Drift Control with Discretionary Stopping for a Diffusion, arXiv:2401.10043v3, §5, second research direction, paragraph proposing infinite-horizon discounting; §2 for the undiscounted model and §3.1 for its variational inequalities. https://arxiv.org/html/2401.10043v3#S5.

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.

Common conventions: Control and stopping conventions for SC-07--SC-08

For the next two entries, retain the diffusion and cost assumptions of Beneš, Gaitsgori and Karatzas, §2: $\mu$ is even, positive and differentiable on $(0,\infty)$; $\sigma$ is even, continuous and bounded away from zero on $(\varepsilon,\infty)$ for every $\varepsilon>0$; $\mu'(x)\sigma(x)^2=A\mu(x)^2$ for $x>0$, with $A\geq0$. If $A=0$, $\sigma$ is bounded; if $A>0$, $\mu'$ is bounded away from zero on each $(\varepsilon,\infty)$. The nonnegative, even functions $k,\psi\in C^2(\R)$ are strictly convex, $\psi(0)=0$, $\psi'(r)\to\infty$ as $r\to\infty$, and $c>0$.

An admissible control/state system is a weak solution, on a filtered probability space carrying a Brownian motion $W$, with progressive real control $u$ and continuous state $X$ satisfying \[ dX_s=u_s\mu(X_s)\,ds+\sigma(X_s)\,dW_s, \quad \int_0^t\left(|u_s\mu(X_s)|+\sigma(X_s)^2\right)ds<\infty \quad\text{a.s. for every }t<\infty. \] Stopping times are taken in the usual augmentation of the state filtration. Infima range over all such weak systems and their associated stopping times; nonnegative costs define extended expectations. This is a single decision maker's optimization problem.