Conditions for prosperity and depression of a stochastic R&D model under regime switching

The stochastic research and development (R&D) model plays an important role in economic growth theories. To explain the growth performance of the economy under regime switching, we first establish sufficient criteria that ensure economic prosperity, nonprosperity and depression in the R&D model disturbed by white and color noise. Then, we determine the threshold between prosperity and depression. Furthermore, we estimate an upper bound of the growth rates of technological progress and capital accumulation in the prosperity case. The results indicate that color noise sensitively impacts the growth performance of the economy in the R&D model. Finally, numerical simulations are conducted to verify our theoretical work.


Introduction
The stochastic research and development (R&D) model has been the center of attention in economic growth theories for a long time, because it can be used to describe the growth rates of technological progress and capital accumulation [1][2][3]. Impacted by the existing economic reality, the uncertainty of the R&D model is affected by many factors, such as the accumulation of knowledge, government intervention, introduction of talent resources, and population fluctuation. The effect of uncertainty on the growth performance of the economy has been studied by several authors [4][5][6][7][8]. For example, Canton [6] analyzed a two-sector model of endogenous growth to reveal the impact of uncertainty on long-run economic growth. He showed that economic growth was higher in the presence of business cycles because people devoted more time to learning activities in an uncertain economic environment. To study endogenous economic growth, the authors of [2,7,8] regarded technological progress as a production process similar to the production of output, and the long-run economic growth rate was determined completely by the population growth rate. Based on various references [2,7,8], Romer [9] fully described the R&D model, interpreted the effectiveness of labor as knowledge, and modeled the determinants of its evolution over time. Wu et al. [4] introduced the uncertainty from population growth and transformed the R&D model of [9] into a simple form via Itô's formula. They computed the sample average of the growth rates of both technology and capital accumulation and proved that the long-run growth rate of the economic system was ultimately bounded in mean. Zhang et al. [10] explored a numerical approximate method that could preserve the positivity of the numerical solution of the R&D model shown in [4] and revealed that the new numerical approximate method was effective and practical.
However, all these works [2,4,[7][8][9][10] assumed that the parameters of the R&D model are all determined constants. In fact, a stochastic R&D system may experience abrupt changes in the structure and parameters. For example, the fixed capital of the listed company is affected by the stock price of the market. Capital accumulation may have optimistic asymptotic behavior when stock prices rise, and capital accumulation may have another asymptotic behavior if stock prices decrease; additionally, the company may enter another development level under the state of talent introduction. Meanwhile, a continuous time Markov chain, namely, color noise, can delineate the system switch from one environmental regime to another in a population system [11][12][13][14][15][16][17]. For instance, Liu et al. [13] showed that an ergodic Markov chain could accurately describe the stochastic phenomena of a population system in practice. Liu et al. [14] added Markovian switching to a stochastic multigroup mutualistic system and investigated the ergodicity property and positive recurrence, which provided a good explanation of some biological phenomena. Yu et al. [15] introduced Markovian switching to the phytoplankton-zooplankton model, they also analyzed the extinction, weak persistence, and nonpersistence in the mean. Zhao et al. [17] developed a stochastic phytoplankton allelopathy model with Markovian switching and showed that the Markovian switching had a great impact on the evolution of the phytoplankton populations.
Inspired by these studies, we introduce Markovian switching into the stochastic R&D model to describe the development trend of the economy under different situations. The switching is memoryless, and the waiting time for the next switch has an exponential distribution. Therefore, the research on the threshold between economic prosperity and depression in the R&D model under regime switching supports our trending analysis and forecasting of the economic environment. Motivated by these features, our natural aims in this paper are as follows: • To get sufficient criteria that maintain economic strong prosperity, weak prosperity, nonprosperity, and depression in the mean.
• To obtain the threshold between stochastic depression and prosperity in the mean.
• To estimate the upper bound of the rates of both technological progress and capital accumulation in the prosperity case.
The rest of this paper is structured as follows. In Sect. 2, we recall the fundamental theory necessary for later discussion and provide sufficient conditions to ensure the ex-  Romer [9] fully described the deterministic R&D model based on [2,7,8]:

Model formulation and preliminaries 2.1 Model formulation
(1) The deterministic R&D model involves four variables: labor (L), capital (K ), technology (A), and output (Y ). There are two sectors, a goods-producing sector, where output is produced, and an R&D sector, where additions to the stock of knowledge (technological progress) are produced. Fraction a L of the labor force is used in the R&D sector, and fraction 1a L is used in the goods-producing sector. Similarly, fraction a K of the capital stock is used in the R&D sector, and the remaining 1a K is used in the goods-producing sector.
Technological progress is regarded as a production process, such as production of output in the model. Both sectors use the full stock of knowledge, where C Y = (1-a K ) 1-α (1a L ) α , is a parameter which measures efficiency in the R&D sector. The population growth of model (1) is regarded as exogenous, andL(t) = nL(t).
Wu [4] introduced uncertainty resulting from population growth, and by means of Itô's formula, the deterministic R&D model (1) can be simplified as a system of stochastic differential equations (SDEs): where η = 1-η, α = 1-α, i.e., they represent the growth rates of technological progress in the R&D sector and the growth rates of capital accumulation in the goods-producing sector, s ∈ (0, 1) means the saving rate, the depreciation rate here of capital K(t) is not considered. Also dL(t) = L(t)(ndt + σ dw(t)), where w(t) is a standard Brownian motion. The features of parameters in model (2) are shown in Table 1.
To describe the stochastic R&D model under different states, we introduce Markovian switching in (2) and assume that there are m regimes, then model (2) obeys: The switching between these m regimes is controlled by a Markov chain and the R&D model (2) with both white and color noise becomes where The analysis of the next sections is focused on model (4).

Preliminary results
For mathematical simplicity, we introduce several notations. Throughout this paper, let (Ω, F, {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (that is, it is right continuous and increasing, while F 0 contains all P-null sets), E denotes the expectation corresponding to P. Let {w(t)} t≥0 be a standard Brownian motion defined on the above complete probability space. For a set A, we denote its indicator function by 1 A , namely 1 A (x) = 1 if x ∈ A and 0 otherwise. Denote by R n + the positive cone in R n , that is, R n [18]. For two real numbers a and b, we use a ∨ b = max{a, b} and a ∧ b = min{a, b}. If A is a vector or matrix, its transpose is denoted by A and the trace norm of A is defined as |A| = trace(A A). If A is an n × n symmetric matrix, we introduce the following definition: x Ax, λ max (A) = sup x∈R n ,|x|=1 x Ax.
It is clear that λ + max (A) ≤ λ max (A) and x Ax ≤ λ + max (A)|x| 2 for any x ∈ R n + [19]. In model (4), taking values in a finite-state S = {1, 2, . . . , m} with the generator Θ = (κ ij ) m×m given by In this paper we note that κ ij > 0 if i = j. As a standing hypothesis, we assume that the Markov chain {ς(t), t ≥ 0} is independent of the Brownian motion w(·) and it is irreducible, which means that the system can switch from any regime to any other [20]. This is equivalent to the condition that for any i, j ∈ S, there are finite numbers i 1 , i 2 , . . . , i i ∈ S such that κ i,i 1 κ i 1 ,i 2 · · · κ i i ,j > 0 and this implies the ergodicity property in view of Markov theory for finite states. If Θ has a trivial eigenvalue, then the algebraic interpretation of irreducbility is rank(Θ) = n -1. Under this assumption, the Markov chain has a unique stationary distribution π = (π 1 , π 2 , . . . , π m ) ∈ R 1×m which can be determined by solving the linear equation πΘ = 0, m k=1 π k = 1 and π k > 0, ∀k ∈ S. The details of the theory have been studied by many authors [11][12][13]21]. For any vector φ = (φ(1), φ(2), . . . , φ(m)) , we note that lim t→∞ . Inspired by Liu and Wang in [16], we also list the following notations: Let (z(t), ς(t)) be the diffusion process described by the following equation: where w(·) and ς(·) are the d-dimensional Brownian motion and the right-continuous Markov chain in the above discussion, respectively. Functions f (·, ·) : For each k ∈ S, and for any twice continuously differentiable function V (·, k), (z(t), ς(t)) has a generator L given as follows [12,21]: For convenience, model (4) can be rewritten in matrix form as where ) .
Proof The proof can be found in Appendix A.
Theorem 2.1 implies that if the returns of technology scale to the capital and technology sectors are decreasing, then the technology and capital show positive long-run growth. This useful property offers us a great opportunity to construct different types of Lyapunov functions to research the asymptotic properties of model (4) in R 2 + in more detail. Next, we attempt to obtain sufficient criteria to ensure economic strong prosperity, weak prosperity, nonprosperity, and depression in the mean. Moreover, we obtain an estimate of the upper bound of economic growth rates.

The threshold between depression and prosperity in the mean
In the R&D model, we are concerned about whether the rates of technological progress and capital accumulation will continue to grow or be stable in the long term. In this section, our aim is to discuss the economic growth rate and give the threshold between depression and prosperity of the system. These are also the main conclusions of this paper.  t 0 y(s) ds > 0 a.s., respectively. Clearly, the condition of strong prosperity in the mean is stronger than that of weak prosperity. However, we define strong prosperity differently from [16] to satisfy the following certifications and actual background.
. From [4], we know that the value of L max can influence the rates of technological progress and capital accumulation in the R&D model with white noise. However, Theorem 3.1 shows that the value of L max is not sufficient to determine the long-run economic trend under regime switching. Therefore, not only L max but also the transition rate κ ij can control the bound of economic growth in the R&D model disturbed by both white and color noise. Ey 2 (s) ds ≤ 2K λ .
Proof By Theorem 2.1, the solution of model (4) will remain in R + × R + × S for all t > 0 a.s., then we define a C 2 -function V : For any positive integer k ≥ (x 2 (0)+y 2 (0)) 1 2 , we define the stopping time ρ k = inf{t ∈ (0, τ e ) : (x 2 (t) + y 2 (t)) 1 2 > k}, where τ e is the explosion time defined in the proof of Theorem 2.1. By the generalized Itô's formula, we can get then integrating both sides from 0 to t ∧ ρ k and taking expectations, we have where , κ i and λ are the same as in Theorem 3.1. Using the Fubini theorem and letting k → ∞, t → ∞, respectively, we get lim sup as desired.
Theorem 3.2 shows that the growth rates of both technological progress and capital accumulation are second moment bounded in the time average sense, and the conclusion is stronger than that of Theorem 3.1.

Proof From Lemma 3.3, we can easily get that
Letting t 0 σ (ς(s))η(ς(s)) dw(s) = M(t), 1 t t 0b (ς(s)) ds =b a.s., we known that M(t) is a martingale which satisfies then, due to the ergodicity of ς(t) and the strong law of large numbers for martingales, we can get so that taking limit superior of both sides of (11) leads to Therefore the desired assertion follows from (13) immediately.

Remark 3.3
In the corresponding deterministic R&D model, the growth rates of both technological progress and capital accumulation are determined entirely by the population growth rate. In the stochastic environment, due to the effect of white noise, if the expected population growth rate n satisfies n < 1-( √ α-√ η) 2 2 σ 2 , then the population growth rate may have little influence on long-run economic growth because the economic growth rate will go to zero. However, the situation changes considerably for the stochastic R&D model under white and color noise. Theorem 3.3 shows that both the ergodicity of the Markov chain ς(t) and the transition rate κ ij from i to j affect the long-run growth rates of technological progress and capital accumulation in the stochastic R&D model under white and color noise.
According to the assumption, we have x(t) + y(t) * > 0.
Remark 3.6 Theorem 3.6 shows that under the condition ofh -1 2 σ 2 (ς(t))[η(ς(t)) ∧ α(ς(t))] 2 * > 0, both technological progress and capital accumulation will be increased significantly in the mean. Clearly, the condition of Theorem 3.6 is stronger than that of Theorem 3.5, which means that if m i=1 π i h(i) is relatively large or the uncertainty from white noise is relatively low, then the economy will be characterized by strong prosperity.

Numerical experiments
In this section, we verify our theory from the previous sections via numerical examples.
Example 4.1 Let ς(t) be a Markov chain on the state space S = {1, 2}, and the generator Θ is given as Θ = -1 1 3 -3 . Fig. 1(a) shows sample paths of the Markov chain, which has two reachable states. The dwell time of the system in these two reachable states depends on the stationary distribution in Fig. 1(b). We can also know that the Markov chain ς(t) spends more time in state 1 and less time in state 2.  Table 2. The Markov chain has the generator Θ = -2 2 1 -1 . It is easy to verify that the parameters satisfy the conditions of Theorem 3.3, according to which we can conclude that the system will be in depression, and the numerical simulation shown in Fig. 2 clearly supports our conclusion.  Table 3. The generator of the Markov chain is Θ = -2 2 5 -5 . According to Theorem 3.6, we can conclude that the model will be in strong prosperity, and the numerical simulation in Fig. 3 displays this phenomenon, as expected.
Example 4.4 Inspired by [23], we obtain the stationary distributions of x(t) and y(t) for three different environmental forcing intensities according to 10,000 numerical simulation runs in Fig. 4. The smooth curves denote the probability density functions of x(t)   The time-series plots of x(t) and y(t) for model (4) with initial value (x 0 , y 0 ) = (0.9, 0.6), and other parameters are taken as in Table 2   Table 3 Parameter values under finite regime switching of model (4) State

Figure 3
The time-series plots of x(t) and y(t) for model (4) with initial (x 0 , y 0 ) = (0.2, 0.8), and other parameters are taken as in Table 3 and y(t). The change in the stationary distributions with increasing σ is illustrated by the distributions displayed in the left (σ = 0.1), middle (σ = 0.4), and right (σ = 0.9) panels in Fig. 4. The parameters of states 1, 2 and 3 are selected according to Table 4.
The results show that increasing the environmental forcing intensity may lead to changes in the mean values and skewness of the distribution for the R&D model. More precisely, the stationary distribution is close to a normal distribution at lower intensity (e.g., σ = 0.1, see the left panel of Fig. 4), but the distribution is positively skewed at higher intensity (e.g., σ = 0.9, see the right panel of Fig. 4).

Concluding remarks
The stochastic R&D model plays an important role in economic growth theories. The asymptotic properties of the R&D model, which is disturbed by white and color noise, is   an open problem. In this paper, we obtain sufficient conditions of economic strong prosperity, weak prosperity, nonprosperity, and depression under regime switching. The most important contribution of this paper is that we precisely express the threshold between prosperity and depression. This means that: • Ifb -σ 2 (ς (t))η 2 (ς (t)) 2 * < 0, then the economy is going into a depression in the long term. • Ifb -σ 2 (ς (t))η 2 (ς (t)) 2 * = 0, then the economy is going into nonprosperity in the long term.
We also find that the growth rates of economic progress and capital accumulation can easily increase in the long term if the Markov chain spends more time in the good states or if the uncertainty coming from white noise is relatively low in the sense that h > 1 2 σ 2 (ς(t))[η(ς(t)) ∧ α(ς(t))] 2 * (see Theorem 3.6). In contrast, if the Markov chain spends more time in the bad states or the uncertainty coming from white noise is relatively large in the sense thatb < σ 2 (ς (t))η 2 (ς (t)) 2 * , it may lead to R&D system depression (see Theorem 3.3). The conclusions of this paper may provide a good explanation for some economic fluctuation phenomena under regime switching. Some questions that need further discussion remain. For instance, it will be useful to extend the R&D model to consider Poisson jumps or time delays. We will make further efforts in this direction in the future.