(Determination of L. by smooth pasting). Consider the...

(Determination of $L$. by smooth pasting). Consider the function $v_L(x)$ in (8.3.11). The first line in formula (8.3.11) implies that the left-hand derivative of $v_L(x)$ at $x=L$ is $v_L\left(L^{-}\right)=-1$. Use the second line in formula (8.3.11) to compute the right-hand derivative $v_L^{\prime}(L+)$. Show that the smooth-pasting condition $$ v_{L_*}^{\prime}\left(L_*-\right)=v_{L *}^{\prime}\left(L_*+\right) $$ is satisfied only by $L *$ given by (8.3.12).

(Determination of $L$. by smooth pasting). Consider the function $v_L(x)$ in (8.3.11). The first line in formula (8.3.11) implies that the left-hand derivative of $v_L(x)$ at $x=L$ is $v_L\left(L^{-}\right)=-1$. Use the second line in formula (8.3.11) to compute the right-hand derivative $v_L^{\prime}(L+)$. Show that the smooth-pasting condition
$$
v_{L_*}^{\prime}\left(L_*-\right)=v_{L *}^{\prime}\left(L_*+\right)
$$
is satisfied only by $L *$ given by (8.3.12).

Key Concepts

Smooth Pasting Condition

This is a condition imposed in optimal stopping and free-boundary problems where the value function is required to be continuously differentiable at the boundary between different regimes. It ensures that the left-hand derivative (approaching the boundary from one side) matches the right-hand derivative (approaching from the other side). Such a match prevents any 'kink' or discontinuity in the derivative, which is indicative of an optimized, smooth transition at the threshold where decisions change.

Free Boundary Problem

In many optimal stopping problems, the decision threshold is not given in advance but must be determined as part of the solution. This leads to free-boundary problems, where one must solve for both the value function and the location of the boundary at which the optimal decision is made. The free boundary is determined by conditions such as value matching and smooth pasting, and represents the optimal transition point.

Optimal Stopping Theory

This area of study focuses on the problem of choosing the best time to take a particular action to maximize expected rewards or minimize expected costs. It involves analyzing the trade-offs in timing decisions, often applying techniques from calculus and stochastic processes. Conditions like value matching and smooth pasting are crucial in identifying the optimal decision rule, including determining thresholds where the optimal stopping rule changes its form.

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