On a nabla fractional boundary value problem with general boundary conditions

1.   Introduction

The models involving entry-exit decisions apply to many situations, as stated in ^[1,2]. Such models may be described simply as follows. A firm decides when to invest in or when to abandon a project that can bring profit. There is plenty of literature on models concerning how to schedule the timing under the assumption that exiting the project does not take time, so we do not list them in detail, but refer to ^{[1,2,3,4,5,6,7,8,9]}. The ideas of entry-exit decisions apply to many concrete issues, for example, relationships of globalization and entrepreneurial entry-exit ^[10], when to invest or expand a start-up firm ^[11], when to enter or exit stock markets ^[12], and how to make a schedule for buying carbon emission rights ^[13].

In practice, the exiting process may take a long time, so ignoring it is not reasonable. For example, Brexit took more than 3 years, from June 23, 2016 to January 31, 2020 (https://www.britannica.com/topic/Brexit). The models ^[14,15] take account of the time, but with equal output rates in the regular production and exit periods. It is an apparent feature that the output rate of the project is usually reduced during the exit period. In this paper, we introduce a parameter to describe the output rate during the exit period and completely discuss the effects of output reduction on entry-exit decisions, under the assumption that the commodity price of the project follows a geometric Brownian motion.

We use the optimal stopping theory to carry out the study and obtain the explicit expressions of the optimal activating time and start time of the exit, which is one of the contributions of the paper. With these explicit expressions in hand, we can carefully analyze the effects of output reduction on entry-exit decisions, which is another contribution of the paper.

If the optimal choice is never to exit the project, the output reduction has no effect on the optimal entry-exit decision.

If the optimal exit is in a finite time, the situation becomes complicated. However, we obtain complete criterions, which determine the effects of output reduction on entry-exit decisions (see Section 4).

We outline the structure of this paper. In Section 2, we describe the model in detail. In Section 3, we determine an optimal entry-exit decision. In Section 4, we discuss the effects of output reduction during exit period on entry-exit decisions. Some conclusions are drawn in Section 5.

2.   The model

Assume that the price process $P$ of one unit product follows

where $\mu\in \mathbb{R}$, $\sigma, \, p > 0$, and $B$ is a one dimensional standard Brownian motion, which denotes uncertainty. In this paper, all times are stopping times w.r.t. the filtration generated by the Brownian motion $B$.

Since involving the construction period leads to complicated calculations and distracts us from analyzing the effects of reduction on entry-exit decisions, and the effects of the construction period have been discussed in ^[3,15], we assume that there is no construction period (it may happen when the firm buys a project). The firm activates the project at time $\tau_I$ with the entry cost $K_I$ and completes the abandonment of the project during the time interval $[\tau_O, \tau_O+\delta]$, with the exit cost valued at $K_O$ at time $\tau_O+\delta$. Without loss of generality, we assume that the firm produces one unit product per unit time during the period $[\tau_I, \tau_O]$ at the marginal cost $C$ and $\alpha$ ($0\leq\alpha\leq1$) unit products per unit time during the time interval of exiting the project $[\tau_O, \tau_O+\delta]$.

To answer the two questions, what time is optimal to activate the project and what time is optimal to start the abandonment procedure, we solve the optimization problem

We call stopping times $\tau_I$ and $\tau_O$ the activating times and start times of the exit, respectively, and we call the function $J$ the maximal expected present value of the project.

3.   An optimal entry-exit decision

If $r\leq \mu$, some straight calculations show us that

Thus, we obtain the following result.

Theorem 3.1. Assume that $r\leq \mu$, then $\tau_I^*$: $= 0$ is an optimal activating time and $\tau_O^*$: $= +\infty$ is an optimal start time of the exit, i.e., the firm should never exit the project. In addition, the function $J$ in (2.2) is given by $J\equiv+\infty$.

In the rest of this section, we assume $r > \mu$.

Taking $\tau_I = \tau_O$: $= 0$ in (2.2), we have

thus, in the remains of this section, we always assume that

to avoid arbitrage opportunities.

Let $\lambda_1$ and $\lambda_2$ be the solutions to the equation

with $\lambda_1 < \lambda_2$, then we have $\lambda_1 < 0$ and $\lambda_2 > 1$.

Theorem 3.2. Assume that $r > \mu$. The following are true:

(ⅰ) If

then $(\tau_I^*, \tau_O^*)$ is a solution to (2.2), where

and $\tau_O^* = +\infty$. Here,

In addition,

where

(ⅱ) If

and

where

then $(\tau_I^*, \tau_O^*)$ is a solution to (2.2), where

and

Here, $p_I$ is the largest solution of the algebraic equation

In addition,

where

and

Remark 3.3. We propose condition (3.1) to eliminate the possibility that the firm enters the project at a trigger price lower than the optimal trigger price of the exit, i.e., the firm enters the project and then immediately decides to exit the project.

In light of [14, Theorems 3.1 and 5.2], we have the following Lemma 3.4, which serves as preparation for the proof of Theorem 3.2.

Lemma 3.4. If $(\tau_1^*, \tau_2^*)$ is a solution to the optimization problem

it is also a solution to (2.2) and $J(x) = \widetilde{J}(x)$, where

and

Proof. By the strong Markov property of the process $\{(s+t, P(t)), t\geq 0\}$, where $s\in\mathbb{R}$, we have

Thus, we need to calculate

The proof is complete. □

Remark 3.5. The proof of Lemma 3.4 has an economic meaning as follows. We first discount the benefit during the abandonment period to time $\tau_O$, then discount this value to time zero.

With the help of Lemma 3.4, we can prove Theorem 3.2.

Proof of Theorem 3.2. (1) By Lemma 3.4, we solve problem (3.2),

where

(2) Assume

We first solve problem (3.4) and then problem (3.3).

For problem (3.4), noting

we see that

which implies

and

Set

Since

the exercise region of problem (3.3) takes the form $[p_I, +\infty)$ for some

The function $H$ satisfies

on the continuation region $(0, p_I)$ and is Lipschitz continuous on $(0, +\infty)$ and $C^1$ continuous at $p_I$. Thus, we get

and $B$ and $p_I$ solve

by which we finish the proof of (ⅰ).

(3) Assume

and (3.1) hold. We again first solve problem (3.4) and then problem (3.3).

Set

A straight calculation shows that

which means the exercise region of problem (3.4) takes the form $(0, p_O]$ for some

The function $G$ satisfies

on the continuation region $(p_O, +\infty)$ and is Lipschitz continuous on $(0, +\infty)$ and $C^1$ continuous at $p_O$. Thus, we get

and $A$ and $p_O$ solve

which implies coefficient $A$ and the optimal exit trigger pricer of (ⅱ).

To solve problem (3.4), we define

In light of (3.1),

thus, the exercise region of problem (3.3) takes the form $[p_I, +\infty)$ for some

The function $H$ satisfies

on the continuation region $(0, p_I)$ and is Lipschitz continuous on $(0, +\infty)$ and $C^1$ continuous at $p_I$. Thus, we get

and $B$ and $p_I$ solve

by which we finish the proof of (ⅱ). □

4.   Discussions

We analyze the effects of output reduction during the exit period on entry-exit decisions.

If $r\leq \mu$, the firm has an optimal time $\tau_I^* = 0$ to activate the project and should never exit the project. Thus, the reduction does not affect entry-exit decisions.

If $r > \mu$ and

the optimal time to activate the project is given by

where

Thus, the reduction does not affect the optimal activating time. As same as the case of $r\leq\mu$, the firm should never exit the project and the reduction does not affect exit decisions.

Assume that $r > \mu$ and

We first analyze the effects of reduction on the optimal start time of the exit. By (ⅱ) of Theorem 3.2, the optimal trigger price is an increasing function of $\alpha$ if

a decreasing function if

and a constant function if

We list some examples to illustrate the analysis. Taking

we have

then the optimal trigger price is an increasing function of $\alpha$. See Figure 1. If we replace $\mu = -0.1$ with $\mu = 0.1$, we have a decreasing function. See Figure 2.

We conclude that:

(1) If

as the reduction increases (i.e., $\alpha$ decreases), the firm will postpone the exit.

(2) If

as the reduction increases, the firm will advance the exit.

(3) If

the reduction does not affect the exit.

To analyze the effects of reduction on the optimal time to activate the project, we define two functions as follows:

and

according to (ⅱ) of Theorem 3.2. Thus, the functions $f$ and $g$ intersect at two points, say, $(p_1, f(p_1))$ and $(p_2, f(p_2))$ with $p_1 < p_2$, and the optimal trigger price $p_I$ of activating the project is given by $p_I = p_2$.

To capture the behavior of $p_I$ as $\alpha$ varies, we only need to examine the behavior of the coefficient $A$ as $\alpha$ varies. Setting

and

after some calculations, we find that:

(1) If $M/N\leq0$, $A$ is increasing on $[0, 1]$ and $p_I$ is decreasing on $[0, 1]$.

(2) If $M/N\geq1$, $A$ is decreasing on $[0, 1]$ and $p_I$ is increasing on $[0, 1]$.

(3) If $0 < M/N < 1$, $A$ is decreasing on $[0, M/N]$ and increasing on $[M/N, 1]$ and $p_I$ is increasing on $[0, M/N]$ and decreasing on $[M/N, 1]$.

Numerical examples help us understand the analysis above.

Figure 3 demonstrates the decreasing of $p_I$ ($r = 0.2$, $\mu = -0.1$, $\sigma = 0.3$, $\delta = 0.6$, $C = 5$, $K_I = 20$, and $K_O = -10$). Figure 4 demonstrates the increasing of $p_I$ ($r = 0.2$, $\mu = 0.1$, $\sigma = 0.3$, $\delta = 2$, $C = 5$, $K_I = 20$, and $K_O = -10$), and Figure 5 demonstrates the non-monotonicity of $p_I$ ($r = 0.2$, $\mu = -0.1$, $\sigma = 0.3$, $\delta = 0.6$, $C = 5$, $K_I = 20$, and $K_O = -10$).

In other words, here are the effects of the reduction on the optimal time of activating the project:

(1) If $M/N\leq0$, as the reduction increases (i.e., $\alpha$ decreases), the firm will postpone activating the project.

(2) If $M/N\geq0$, as the reduction increases, the firm will advance activating the project.

(3) If $0 < M/N < 1$, as the reduction increases, the firm will postpone activating the project then advance activating the project.

5.   Conclusions

There is an apparent phenomenon that firms usually reduce their output rate during stopping the regular production of a project. We intend to analyze the effects of reduction on the optimal entry-exit decision. Since the papers ^[3,15] have investigated the effects of the construction period on entry-exit decisions, we assume that the project has been constructed and the production immediately starts for concentrating on the study of the effects of reduction.

We introduce the output rate into the models ^[14,15] and describe the problem using the optimal stopping theory, obtaining explicit solutions in Theorems 3.1 and 3.2. These explicit solutions help us in discovering the effects of reduction on the optimal entry-exit decision.

We carefully examine the effects of reduction. According the analysis listed in Section 4, we come to the effects of reduction as follows. If the firm should never exit the project to obtain the maximal profit, the reduction does not affect the optimal entry-exit decision. However, if the firm exits the project in finite time, the effects show in a complicated manner. We study the effects analytically and numerically, and provide the relations of the parameters involved in the model that can determine the effects.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Many thanks are due to the editors and reviewers for their constructive suggestions and valuable comments. This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. N2023034).

Conflict of interest

No potential conflict of interest exists regarding the publication of this paper.