Author

Maria B. Chiarolla

Date Updated

2022/09/20

Category

math.OC

Date Published

2022/09/20

Date Retrieved

2022/09/21

Description

This paper studies the investment exercise boundary erasing in a stochastic,
continuous time capacity expansion problem with irreversible investment on the
finite time interval $[0, T]$ and a state dependent scrap value associated with
the production facility at the finite horizon $T$. The capacity process is a
time-inhomogeneous diffusion in which a monotone nondecreasing, possibly
singular, process representing the cumulative investment enters additively. The
levels of capacity, employment and operating capital contribute to the firm's
production and are optimally chosen in order to maximize the expected total
discounted profits. Two different approaches are employed to study and
characterize the boundary. From one side, some first order condition are solved
by using the Bank and El Karoui Representation Theorem, and that sheds further
light on the connection between the threshold which the optimal policy of the
singular stochastic control problem activates at and the optional solution of
Representation Theorem. Its application in the presence of the scrap value is
new. It is accomplished by a suitable devise to overcome the difficulties due
to the presence of a non integral term in the maximizing functional. The
optimal investment process is shown to become active at the so-called "base
capacity" level, given as the unique solution of an integral equation. On the
other hand, when the coefficients of the uncontrolled capacity process are
deterministic, the optimal stopping problem classically associated to the
original capacity problem is resumed and some essential properties of the
investment exercise boundary are obtained. The optimal investment process is
proved to be continuous. Unifying approaches and views, the exercise boundary
is shown to coincide with the base capacity, hence it is characterized by an
integral equation not requiring any a priori regularity.

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URL

https://arxiv.org/abs/2209.09878

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