Wong-Zakai approximations and long term behavior of second order non-autonomous stochastic lattice dynamical systems with additive noise

Xintao Li; Xintao Li

doi:10.3934/math.2022425

AIMS Mathematics

2022, Volume 7, Issue 5: 7569-7594. doi: 10.3934/math.2022425

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Research article

Wong-Zakai approximations and long term behavior of second order non-autonomous stochastic lattice dynamical systems with additive noise

Xintao Li ^,

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received: 04 November 2021 Revised: 18 January 2022 Accepted: 24 January 2022 Published: 15 February 2022
MSC : 37L30, 37L55, 37L60

In this article, we investigate the Wong-Zakai approximations of a class of second order non-autonomous stochastic lattice systems with additive white noise. We first prove the existence and uniqueness of tempered pullback random attractors for the original stochastic system and its Wong-Zakai approximation. Then, we establish the upper semicontinuity of these attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero.

Keywords:

Citation: Xintao Li. Wong-Zakai approximations and long term behavior of second order non-autonomous stochastic lattice dynamical systems with additive noise[J]. AIMS Mathematics, 2022, 7(5): 7569-7594. doi: 10.3934/math.2022425

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Abstract

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

$\begin{equation} \mathrm{d}P(t) = \mu P(t)\mathrm{d}t+\sigma P(t)\mathrm{d}B(t) \;{\rm{and}}\; P(0) = p , \end{equation}$

(2.1)

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

$\begin{equation} \begin{aligned} J(p) = &\sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t+\alpha\int_{\tau_O}^{\tau_O+\delta}\exp(-r t)(P(t)-C)\mathrm{d}t\\ &-\exp(-r\tau_I)K_I-\exp(-r(\tau_O+\delta))K_O\bigg]. \end{aligned} \end{equation}$

(2.2)

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

$\mathbb{E}^p\left[\int_{0}^{+\infty}\exp(-r t)(P(t)-C)\mathrm{d}t\right] = +\infty.$

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

$J(p)\geq -K_I+\frac{\alpha(1-\exp((\mu-r)\delta))}{r-\mu}p-\exp(-r\delta)K_O-\alpha\frac{C}{r}(1-\exp(-r\delta)),$

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

$rK_I+\exp(-r\delta)rK_O+\alpha(1-\exp(-r\delta))C\geq0$

to avoid arbitrage opportunities.

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

$r-\mu\lambda-\frac{1}{2}\sigma^2\lambda(\lambda-1) = 0$

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

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)\leq0,$

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

$\tau_I^* = \inf\{t: t > 0, P(t)\geq p_I\}$

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

$p_I = \frac{\lambda_2}{\lambda_2-1}(r-\mu)\left(\frac{C}{r}+K_I\right).$

In addition,

$J(p) = \left\{ \begin{aligned} &B p^{\lambda_2}, \quad \mathit{\text{if}}\;\,p < p_I,\\ &\frac{p}{r-\mu}-\frac{C}{r}-K_I,\;\;\mathit{\text{if}}\;\, p\geq p_I, \end{aligned} \right.$

where

$B = \frac{{p_I}^{1-\lambda_2}}{\lambda_2(r-\mu)}.$

(ⅱ) If

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta) > 0$

and

$\begin{equation} \alpha(1-\exp((\mu-r)\delta))p_O-\alpha C(1-\exp(-r\delta))-\exp(-r\delta)rK_O-rK_I\leq 0, \end{equation}$

(3.1)

where

$p_O = \frac{\lambda_1}{\lambda_1-1}\frac{r-\mu}{1-\alpha+\alpha\exp((\mu-r)\delta)}\left((1-\alpha)\frac{C}{r}+\exp(-r\delta)\bigg(\alpha\frac{C}{r}-K_O\bigg)\right),$

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

$\tau_I^* = \inf\{t: t > 0, P(t)\geq p_I\}$

and

$\tau_O^* = \inf\{t: t > \tau_I^*, P(t)\leq p_O\}.$

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

$A(\lambda_2-\lambda_1)p_I^{\lambda_1}+\frac{(\lambda_2-1)}{r-\mu}p_I-\lambda_2\left(\frac{C}{r}+K_I\right) = 0.$

In addition,

$J(p) = \left\{ \begin{aligned} &B p^{\lambda_2}, \quad \mathit{\text{if}}\;\, p < p_I,\\ &Ap^{\lambda_1}+\frac{p}{r-\mu}-\frac{C}{r}-K_I,\;\;\mathit{\text{if}}\;\, p\geq p_I, \end{aligned} \right.$

where

$A = \frac{1-\alpha+\alpha\exp((\mu-r)\delta)}{\lambda_1(\mu-r)}{p_O}^{1-\lambda_1}$

and

$B = \lambda_1\lambda_2^{-1}Ap_I^{\lambda_1-\lambda_2}+\frac{{p_I}^{1-\lambda_2}}{\lambda_2(r-\mu)}.$

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

$\begin{equation} \widetilde{J}(p): = \sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t -\exp(-r\tau_I)K_I-\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\bigg], \end{equation}$

(3.2)

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

$l_1: = \frac{\alpha(1-\exp((\mu-r)\delta))}{\mu-r}$

and

$l_0: = \alpha\frac{C}{r}(1-\exp(-r\delta))+\exp(-r\delta)K_O.$

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

$\begin{aligned} &\mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t+\alpha\int_{\tau_O}^{\tau_O+\delta}\exp(-r t)(P(t)-C)\mathrm{d}t -\exp(-r\tau_I)K_I-\exp(-r(\tau_O+\delta))K_O\bigg]\\ & = \mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t+\alpha\exp(-r\tau_O)\mathbb{E}^{P(\tau_O)}\bigg[\int_0^{\delta}\exp(-r t)(P(t)-C)\mathrm{d}t\bigg]\\ &\quad\ -\exp(-r\tau_I)K_I-\exp(-r(\tau_O+\delta))K_O\bigg]. \end{aligned}$

Thus, we need to calculate

$\begin{aligned} &\alpha\mathbb{E}^{P(\tau_O)}\bigg[\int_0^{\delta}\exp(-r t)(P(t)-C)\mathrm{d}t\bigg]-\exp(-r\delta)K_O\\ & = \alpha\int_0^{\delta}\exp(-r t)(\exp(\mu t)P(\tau_O)-C)\mathrm{d}t-\exp(-r\delta)K_O\\ & = -l_1P(\tau_O)-l_0. \end{aligned}$

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),

$\begin{align} &\sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t-\exp(-r\tau_I)K_I-\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\bigg]\\ & = \sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\bigg[\exp(-r\tau_I)\int_{0}^{\tau_O-\tau_I}\exp(-r t)(P(t+\tau_I)-C)\mathrm{d}t\\&\quad-\exp(-r\tau_I)K_I-\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\bigg]\\ & = \sup\limits_{\tau_I}\mathbb{E}^p\bigg[\exp(-r\tau_I)(G(P(\tau_I))-K_I)\bigg]\\ & = :H(p), \end{align}$

(3.3)

where

$\begin{equation} G(p): = \sup\limits_{\tau_O}\mathbb{E}^p\bigg[\int_{0}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t -\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\bigg]. \end{equation}$

(3.4)

(2) Assume

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)\leq0 .$

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

For problem (3.4), noting

$\mathbb{E}^p\bigg[\int_0^\infty\exp(-rt)(P(t)-C)\mathrm{d}t\bigg] = \frac{p}{r-\mu}-\frac{C}{r},$

we see that

$\mathbb{E}^p\bigg[\int_0^\infty\exp(-rt)(P(t)-C)\mathrm{d}t\bigg]\geq -l_1p-l_0,$

which implies

$\tau_O^* = +\infty$

and

$G(p) = \frac{p}{r-\mu}-\frac{C}{r}.$

Set

$h(p): = \frac{p}{r-\mu}-\frac{C}{r}-K_I.$

Since

$\{p:p > 0\;\mathrm{and}\;rh(p)-\mu ph'(p)\geq 0\} = \{p:p\geq C+rK_I\},$

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

$p_I\geq C+rK_I.$

The function $H$ satisfies

$rH-\mu pH'-\frac{1}{2}\sigma^2p^2H'' = 0$

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

$H(p) = Bp^{\lambda_2}$

and $B$ and $p_I$ solve

$\left\{ \begin{aligned} &B p_I^{\lambda_2} = \frac{p_I}{r-\mu}-\frac{C}{r}-K_I,\\ &\lambda_2Bp_I^{\lambda_2-1} = \frac{1}{r-\mu}, \end{aligned} \right.$

by which we finish the proof of (ⅰ).

(3) Assume

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta) > 0$

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

Set

$g(p): = -l_1p-l_0.$

A straight calculation shows that

$\begin{aligned} \{p:p > 0\;\mathrm{and}\;rg-\mu p g'(p)-p+C\geq0\} = \bigg(0,\frac{(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)}{1-\alpha(1-\exp((\mu-r)\delta))}\bigg), \end{aligned}$

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

$p_O\leq \frac{(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)}{1-\alpha(1-\exp((\mu-r)\delta))}.$

The function $G$ satisfies

$rG-\mu pG'-\frac{1}{2}\sigma^2p^2G''-p+C = 0$

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

$G(p) = Ap^{\lambda_1}+\frac{p}{r-\mu}-\frac{C}{r},$

and $A$ and $p_O$ solve

$\left\{ \begin{aligned} Ap_O^{\lambda_1}+\frac{p_O}{r-\mu}-\frac{C}{r}& = -l_1p_O-l_0,\\ \lambda_1Ap_O^{\lambda_1-1}+\frac{1}{r-\mu}& = -l_1, \end{aligned} \right.$

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

To solve problem (3.4), we define

$h(p): = G(p)-K_I.$

In light of (3.1),

$\{p:p > 0\;\mathrm{and}\;rh(p)-\mu ph'(p)\geq 0\} = \{p:p\geq C+rK_I\},$

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

$p_I\geq C+rK_I.$

The function $H$ satisfies

$rH-\mu pH'-\frac{1}{2}\sigma^2p^2H'' = 0$

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

$H(p) = Bp^{\lambda_2},$

and $B$ and $p_I$ solve

$\left\{ \begin{aligned} B p_I^{\lambda_2}& = Ap_I^{\lambda_1}+\frac{p_I}{r-\mu}-\frac{C}{r}-K_I,\\ \lambda_2Bp_I^{\lambda_2-1}& = \lambda_1Ap_I^{\lambda_1-1}+\frac{1}{r-\mu}, \end{aligned} \right.$

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

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)\leq0,$

the optimal time to activate the project is given by

$\tau_I^* = \inf\{t: t > 0, P(t)\geq p_I\},$

where

$p_I = \frac{\lambda_2}{\lambda_2-1}(r-\mu)\left(\frac{C}{r}+K_I\right).$

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

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta) > 0.$

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

$\exp(-\mu\delta)(C-rK_O) > C-\exp(-r\delta)rK_O,$

a decreasing function if

$\exp(-\mu\delta)(C-rK_O) < C-\exp(-r\delta)rK_O,$

and a constant function if

$\exp(-\mu\delta)(C-rK_O) = C-\exp(-r\delta)rK_O.$

We list some examples to illustrate the analysis. Taking

$r = 0.2,\; \; \; \mu = -0.1,\; \; \; \sigma = 0.3,\; \; \; \delta = 2,\; \; \; C = 5,\; \; \; K_I = 20,\; \; \text{and}\; \; K_O = -10,$

we have

$\exp(-\mu\delta)(C-rK_O) > C-\exp(-r\delta)rK_O,$

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

Figure 1.

$\alpha$ -

$p_O$ -increasing.

DownLoad: Full-Size Img PowerPoint

Figure 2.

$\alpha$ -

$p_O$ -decreasing.

DownLoad: Full-Size Img PowerPoint

We conclude that:

(1) If

$\exp(-\mu\delta)(C-rK_O) > C-\exp(-r\delta)rK_O,$

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

(2) If

$\exp(-\mu\delta)(C-rK_O) < C-\exp(-r\delta)rK_O,$

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

(3) If

$\exp(-\mu\delta)(C-rK_O) = C-\exp(-r\delta)rK_O,$

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:

$f(p): = A(\lambda_2-\lambda_1)p^{\lambda_1}$

and

$g(p): = -\frac{(\lambda_2-1)}{r-\mu}p+\lambda_2\left(\frac{C}{r}+K_I\right)$

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

$\begin{align*} M: = &(C-\exp(-r\delta)rK_O)(1-\exp((\mu-r)\delta))\\&+(1-\lambda_1)\exp(-r\delta)(C-rK_O-C\exp(\mu\delta)+\exp((\mu-r)\delta)rK_O) \end{align*}$

and

$N: = -C(1-\exp(-r\delta))(1-\exp((\mu-r)\delta)),$

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.

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

Figure 3.

$\alpha$ -

$p_I$ -decreasing.

DownLoad: Full-Size Img PowerPoint

Figure 4.

$\alpha$ -

$p_I$ -increasing.

DownLoad: Full-Size Img PowerPoint

Figure 5.

$\alpha$ -

$p_I$ -non-monotonicity.

DownLoad: Full-Size Img PowerPoint

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.

References

[1]	L. Arnold, Random dynamical systems, 1 Eds., Berlin: Springer, 1998. http://dx.doi.org/10.1007/978-3-662-12878-7
[2]	P. W. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1–21. http://dx.doi.org/10.1142/S0219493706001621 doi: 10.1142/S0219493706001621
[3]	Z. Brzeźniak, U. Manna, D. Mukherjee, Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations, J. Differ. Equations, 267 (2019), 776–825. http://dx.doi.org/10.1016/j.jde.2019.01.025 doi: 10.1016/j.jde.2019.01.025
[4]	T. L. Carrol, L. M. Pecora, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821–824. http://dx.doi.org/10.1103/PhysRevLett.64.821 doi: 10.1103/PhysRevLett.64.821
[5]	P. Chen, R. Wang, X. Zhang, Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains, B. Sci. Math., 173 (2021), 103071. http://dx.doi.org/10.1016/j.bulsci.2021.103071 doi: 10.1016/j.bulsci.2021.103071
[6]	C. Cheng, Z. Feng, Y. Su, Global stability of traveling wave fronts for a reaction-diffusion system with a quiescent stage on a one-dimensional spatial lattice, Appl. Anal., 97 (2018), 2920–2940. http://dx.doi.org/10.1080/00036811.2017.1395864 doi: 10.1080/00036811.2017.1395864
[7]	L. O. Chua, T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147–156. http://dx.doi.org/10.1109/81.222795
[8]	X. Ding, J. Jiang, Random attractors for stochastic retarded lattice dynamical systems, Abstract. Appl. Anal., 2012 (2012), 409282. http://dx.doi.org/10.1155/2012/409282 doi: 10.1155/2012/409282
[9]	A. Gu, Asymptotic behavior of random lattice dynamical systems and their wong-zakai approximations, Discrete Contin. Dyn. Syst. B, 24 (2019), 5737–5767. http://dx.doi.org/10.3934/dcdsb.2019104 doi: 10.3934/dcdsb.2019104
[10]	A. Gu, K. Lu, B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185–218. http://dx.doi.org/10.3934/dcds.2019008 doi: 10.3934/dcds.2019008
[11]	A. Gu, B. Guo, B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, Discrete Contin. Dyn. Syst. B, 25 (2020), 2495–2532. http://dx.doi.org/10.3934/dcdsb.2020020 doi: 10.3934/dcdsb.2020020
[12]	J. Guo, C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differ. Equations, 260 (2016), 1445–1455. http://dx.doi.org/10.1016/j.jde.2015.09.036 doi: 10.1016/j.jde.2015.09.036
[13]	Z. Han, S. Zhou, Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh-Nagumo lattice systems with quasi-periodic forces and multiplicative noise, Stoch. Dyn., 20 (2020), 2050036. http://dx.doi.org/10.1142/S0219493720500367 doi: 10.1142/S0219493720500367
[14]	R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113–163. http://dx.doi.org/10.1007/BF01192578 doi: 10.1007/BF01192578
[15]	D. Li, L. Shi, Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg-Landau equations with time-varying delays in the delay, J. Differ. Equ. Appl., 24 (2018), 872–897. http://dx.doi.org/10.1080/10236198.2018.1437913 doi: 10.1080/10236198.2018.1437913
[16]	D. Li, L. Shi, X. Wang, Long term behavior of stochastic discrete complex ginzburg-landau equations with time delays in weighted spaces, Discrete Contin. Dyn. Syst. B, 24 (2019), 5121–5148. http://dx.doi.org/10.3934/dcdsb.2019046 doi: 10.3934/dcdsb.2019046
[17]	K. Lu, B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat., 31 (2019), 1341–1371. http://dx.doi.org/10.1007/s10884-017-9626-y doi: 10.1007/s10884-017-9626-y
[18]	K. Lu, Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differ. Equations, 251 (2011), 2853–2895. http://dx.doi.org/10.1016/j.jde.2011.05.032 doi: 10.1016/j.jde.2011.05.032
[19]	J. Malletparet, S. Chow, Pattern formation and spatial chaos in lattice dynamical systems I, IEEE Transactions on Circuits Systems I Fundamental Theory Applications, 42 (2002), 746–751. http://dx.doi.org/10.1109/81.473583 doi: 10.1109/81.473583
[20]	U. Manna, D. Mukherjee, A. A. Panda, Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations with anisotropy energy, J. Math. Anal. Appl., 480 (2019), 123384. http://dx.doi.org/10.1016/j.jmaa.2019.123384 doi: 10.1016/j.jmaa.2019.123384
[21]	L. She, R. Wang, Regularity, forward-compactness and measurability of attractors for non-autonomous stochastic lattice systems, J. Math. Anal. Appl., 479 (2019), 2007–2031. http://dx.doi.org/10.1016/j.jmaa.2019.07.038 doi: 10.1016/j.jmaa.2019.07.038
[22]	J. Shen, K. Lu, W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differ. Equations, 255 (2013), 4185–4225. http://dx.doi.org/10.1016/j.jde.2013.08.003 doi: 10.1016/j.jde.2013.08.003
[23]	H. Su, S. Zhou, L. Wu, Random exponential attractor for second-order nonautonomous stochastic lattice systems with multiplicative white noise, Stoch. Dyn., 19 (2019), 1950044. http://dx.doi.org/10.1142/S0219493719500448 doi: 10.1142/S0219493719500448
[24]	B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009. http://dx.doi.org/10.1142/S0219493714500099 doi: 10.1142/S0219493714500099
[25]	X. Wang, D. Li, J. Shen, Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise, Discrete Contin. Dyn. Syst. B, 26 (2021), 2829–2855. http://dx.doi.org/10.3934/dcdsb.2020207 doi: 10.3934/dcdsb.2020207
[26]	R. Wang, Y. Li, B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091–4126. http://dx.doi.org/10.3934/dcds.2019165 doi: 10.3934/dcds.2019165
[27]	X. Wang, S. Li, D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear. Anal. Theor., 72 (2010), 483–494. http://dx.doi.org/10.1016/j.na.2009.06.094 doi: 10.1016/j.na.2009.06.094
[28]	X. Wang, K. Lu, B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equations, 264 (2018), 378–424. http://dx.doi.org/10.1016/j.jde.2017.09.006 doi: 10.1016/j.jde.2017.09.006
[29]	X. Wang, K. Lu, B. Wang, Stationary approximations of stochastic wave equations on unbounded domains with critical exponents, J. Math. Phys., 62 (2021) 092702. http://dx.doi.org/10.1063/5.0011987
[30]	X. Wang, J. Shen, K. Lu, B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differ. Equations, 280 (2021), 477–516. http://dx.doi.org/10.1016/j.jde.2021.01.026 doi: 10.1016/j.jde.2021.01.026
[31]	R. Wang, L. Shi, B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $R^N$ , Nonlinearity, 32 (2019), 4524–4556. http://dx.doi.org/10.1088/1361-6544/ab32d7 doi: 10.1088/1361-6544/ab32d7
[32]	R. Wang, B. Wang, Random dynamics of P-Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stoch. Proc. Appl., 130 (2020), 7431–7462. http://dx.doi.org/10.1016/j.spa.2020.08.002 doi: 10.1016/j.spa.2020.08.002
[33]	R. L. Winalow, A. L. Kimball, A. Varghese, Simulating cartidiac sinus and atrial network dynamics on connection machine, Physica D, 64 (1993), 281–298. http://dx.doi.org/10.1016/0167-2789(93)90260-8 doi: 10.1016/0167-2789(93)90260-8
[34]	C. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dyn. Differ. Equ., 28 (2016), 317–338. http://dx.doi.org/10.1007/s10884-016-9524-8 doi: 10.1007/s10884-016-9524-8
[35]	X. Xiang, S. Zhou, Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term, J. Differ. Equ. Appl., 22 (2016), 235–252. http://dx.doi.org/10.1080/10236198.2015.1080694 doi: 10.1080/10236198.2015.1080694
[36]	L. Xu, W. Yan, Stochastic FitzHugh-Nagumo systems with delay, Taiwan. J. Math., 16 (2012), 1079–1103. http://dx.doi.org/10.11650/twjm/1500406680 doi: 10.11650/twjm/1500406680
[37]	W. Yan, Y. Li, S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702. http://dx.doi.org/10.1063/1.3319566 doi: 10.1063/1.3319566
[38]	C. Zhang, L. Zhao, The attractors for 2nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Sys., 37 (2017), 575–590. http://dx.doi.org/10.3934/dcds.2017023 doi: 10.3934/dcds.2017023
[39]	S. Zhou, Attractors for second-order lattice dynamical systems with damping, J. Math. Phys., 43 (2002), 452–465. http://dx.doi.org/10.1063/1.1418719 doi: 10.1063/1.1418719
[40]	S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differ. Equations, 263 (2020), 2247–2279. http://dx.doi.org/10.1016/j.jde.2017.03.044 doi: 10.1016/j.jde.2017.03.044

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Wong-Zakai approximations and long term behavior of second order non-autonomous stochastic lattice dynamical systems with additive noise

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1. Introduction

2. The model

3. An optimal entry-exit decision

4. Discussions

5. Conclusions

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Use of AI tools declaration

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Wong-Zakai approximations and long term behavior of second order non-autonomous stochastic lattice dynamical systems with additive noise

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Abstract

1. Introduction

2. The model

3. An optimal entry-exit decision

4. Discussions

5. Conclusions

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Acknowledgments

Conflict of interest

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Abstract

1. Introduction

2. The model

3. An optimal entry-exit decision

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5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

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