The one-shot problem: Solution to an open question of finite-fuel singular control with discretionary stopping

Abstract

We introduce a novel 'one-shot' solution technique resolving an open problem (Karatzas et al., Finite-fuel singular control with discretionary stopping, Stochastics 71:1-2 (2000)). Unexpectedly given the convexity of the latter problem, its waiting region is not necessarily connected. Along a typical sample path, the state process may even spend positive time in both of its connected components. The analysis reveals more generally that when fuel is limited, contrary to intuition, the solution without fuel is not necessarily indicative of the solution for small amounts of fuel. To resolve this, we recommend solving the `one-shot' problem, which is one of optimal stopping, prior to employing the usual `guess and verify' solution approach.

Figures (9)

• Figure 1: Moving boundaries of the control problem when $\lambda \geq \alpha \delta$ (obtained in KOWZ00). In the stopping region $S$ the process is absorbed. The direction of control is south-west (grey lines). In the action region $A$, fuel is expended to drive the process in this direction towards its boundary $\partial A$, where it is then absorbed. There is no waiting region.
• Figure 2: Moving boundaries of the control problem when $\lambda \in [\lambda^\dagger, \alpha \delta)$, obtained in Section \ref{['sec:cpos']}. When the fuel level is $C_t>0$, the absorbing boundary is located at $X_t = F(C_t) < \frac{1}{2\delta}$. However when the fuel level $C_t = 0$, the absorbing boundary is located at $X_t = F(0) = f_0 > \frac{1}{2\delta}$, i.e. $F$ is discontinuous. The right boundary $G$ is reflecting for $c>\overline c$ and repelling for $c \in (0,\overline c]$, and $W_1$ is a waiting region. The nature of the stopping region $S$ and action region $A$ are as in Figure \ref{['fig:boundaries0']}.
• Figure 3: Moving boundaries of the control problem when $\lambda \in (\lambda^*, \lambda^\dagger)$, obtained in Section \ref{['sec:newsol']}. Whereas the boundaries $F$ and $G$ border a component $W_1$ of the waiting region as in Figure \ref{['fig:boundaries4']}, now an additional repelling boundary $\overline F$ and reflecting boundary $\overline G$ create a separate waiting region $W_2$. The nature of the stopping region $S$ and action region $A$ are as in Figure \ref{['fig:boundaries0']}.
• Figure 4: Moving boundaries of the control problem when $f_0 \leq \frac{1}{2\delta}$ and $\lambda \in (\lambda_*,\lambda^*]$ (obtained in KOWZ00). In this case the absorbing boundary $F$ is continuous at $0$. The nature of the stopping region $S$ and action region $A$ are as in Figure \ref{['fig:boundaries0']}.
• Figure 5: A geometric representation of the optimal stopping problem $V_0$ of \ref{['eq:protonofuel']} when $\lambda \in (\lambda^*, \alpha \delta)$, showing the transformed obstacle $H_l$ on $[1,\infty)$ (solid curve) and its greatest non-positive convex minorant $W$ (dotted curve). The two curves coincide for $y \in [1,\Psi(f_0)]$ and the tangency point is shown (square marker). For convenience the natural scale $x=\Psi^{-1}(y)$ is also given, and the stopping region in the natural scale is $[0,f_0]$.

Theorems & Definitions (55)

• Remark 1.1
• Definition 1.2
• Remark 3.1
• Proposition 3.2: Dayanik2003
• Lemma 3.3: cf. Figure \ref{['fig:1']}
• Theorem 3.4
• proof
• Lemma 3.5
• Lemma 3.6
• Definition 4.1