Uncertain Stochastic Hybrid Age-Dependent Population Equation Based on Subadditive Measure: Existence, Uniqueness and Exponential Stability

Abstract

The existing literature lacks a study on age-dependent population equations based on subadditive measures. In this paper, we propose a hybrid age-dependent population dynamic system (referred to as APDS) that incorporates uncertain random perturbations driven by both the well-known Wiener process and the Liu process associated with belief degree, which have similar symmetry in terms of form. Firstly, we redefine the Liu integral in a mean square sense and then extend Liu’s lemma and the Itô-Liu formula. We then utilize the extensions of the Itô-Liu formula, Barkholder-Davis-Gundy (BDG) inequality, the Liu’s lemma, the Gronwall’s lemma and the symmetric nature of calculus itself to establish the uniqueness of a strong solution for the hybrid APDS. Additionally, we prove the existence of the hybrid APDS by combining the proof of uniqueness with some important lemmas. Finally, under appropriate assumptions, we demonstrate the exponential stability of the hybrid system.

1. Introduction

The impact of environmental noise on population systems has long been recognized. In previous studies on stochastic population systems [1,2], the focus was primarily on modeling changes caused by random factors during death. However, interactions between species or random fluctuations in the environment were not considered. In 2004, Zhang et al. [3] introduced the Wiener process to the age-dependent model without diffusion and demonstrated the existence, uniqueness, and exponential stability of the stochastic APDS. Subsequently, in 2005, Zhang et al. [4] conducted numerical analysis on the stochastic APDS. In 2006, Li et al. [5] performed numerical simulations of the model using Euler’s approximate solution and compared it with the analysis solution without perturbation and the alternative explicit solution. For more models on using the Euler-Maruyama approximation method to deal with stochastic differential systems driven by Wiener process, please refer to Refs. [6,7], Moreover, in recent decades, a significant and expanding body of literature has emerged, investigating models and numerical algorithms for stochastic age-dependent population equations, such as [8,9].

In [3,4,5], the stochastic age-dependent model without diffusion is as follows:

$$\left\{\begin{array}{cc}{d}_{k}N=-\frac{\partial N}{\partial \theta }dk-\eta (k,\theta )Ndk+f(k,N)dk+g(k,N)d{W}_k,\hfill & \hfill inU=(0,\Theta )\times (0,T),\\ N(0,\theta )=N_0(\theta ),\hfill & \hfill in[0,\Theta ],\\ N(k,0)={\int }_0^{\Theta }\zeta (k,\theta )N(k,\theta )d\theta ,\hfill & \hfill in[0,T],\end{array}\right.$$

(1)

where $\theta \in (0,\Theta )$ stands for age, $k\in (0,T)$ stands for time, $N(k,\theta )$ stands for the population density of age $\theta$ at time k, $\eta (k,\theta )$ stands for the mortality rate of age $\theta$ at time k, $\zeta (k,\theta )$ stands for the fertility rate of females of age $\theta$ at time k, $f(k,N)$ stands for influence of external environment for population system. $d_{k}N$ is the differential of N related to time k, that is, $d_{k}N=\frac{\partial N}{\partial k}dk$, $\dot{W}(k)$ is Wiener process.

Uncertainty theory [10] provides a useful account of how to deal with uncertain disturbance related to belief degrees of indeterminate phenomena. In order to describe the evolution of an uncertain phenomenon related to belief degrees, the Liu process [11] was first applied in uncertain differential equations (UDEs for short) [12]. After that, Yao et al. [13] first provided a theorem on Lipschitz continuity of process, based on which, it gave a sufficient condition for an uncertain differential equation being stable. Furthermore, Yao and Chen [14] first developed the Yao-Chen formula to simulate UDEs. In recent years, they have been successfully applied to many directions such as differential game [15,16], partial differential equation [17] and age-dependent population [18], fractional differential equation related to symmetry [19] and so on. The definition of Liu process $C_k$ is as follows

(a) $C_0=0$ and almost all sample paths are Lipschitz continuous,

(b) $C_k$ has stationary and independent increments,

(c) every increment $C_{r+k}-C_r$ is a normal uncertain variable with expected value 0 and variance $k^2$, whose uncertainty distribution is

$$\Phi (x)={\left(1+exp\left(\frac{-\pi x}{\sqrt{3}k}\right)\right)}^{-1},x\in \mathbb{R},$$

and inverse distribution with belief degree $\alpha$$(\alpha \in (0,1))$ is

$${\Phi }_k^{-1}(\alpha )=\frac{\sqrt{3}\Delta k}{-\pi }ln(\frac{1}{\alpha }-1).$$

Randomness and uncertainty are frequently prescribed for indeterminacy and have been given more and more attention. For a complex hybrid system, recently, there has been new interest in chance space [20], which can effectively deal with the issue of randomness and uncertainty. Uncertain stochastic process is a sequence of uncertain random variables indexed by time. With demand of practical issues need, recently investigators have examined the effects of uncertain random elements on the differential equation system. In 2014, similar to the analysis method of the fuzzy random process, Fei [21] first considered uncertain stochastic differential equations (USDEs), proved the Itô-Liu formula and applied it to control systems with Markovian switching. In the same year, Fei [22] proved the existence and uniqueness theorems of backward USDEs. Of course, there are many papers about hybrid systems, such as [23,24,25].

However, none of the previous studies have attempted to investigate the age-dependent population dynamic with uncertainty related to belief degree and randomness at the same time. Consequently, this paper attempts to establish the theoretical research about the hybrid age-dependent population dynamic. Suppose that $-\eta (k,\theta )N+f(k,N)$ is uncertain stochastically perturbed, with

$$-\eta (k,\theta )N+f(k,N)\to -\eta (k,\theta )N+f(k,N)+g(k,N){\dot{W}}_k+h(k,N){\dot{C}}_k,$$

where ${\dot{W}}_k$ is a Wiener process and ${\dot{C}}_k$ is a Liu process, both of which are environmental disturbances. Then, we use the following Itô-Liu equation to describe environmentally perturbed population system.

$$\left\{\begin{array}{cc}{d}_{k}N=-\frac{\partial N}{\partial \theta }dk-\eta (k,\theta )Ndk+f(k,N)dk+g(k,N)d{W}_k+h(k,N)d{C}_k,\hfill & \hfill inU=(0,\Theta )\times (0,T),\\ N(0,\theta )=N_0(\theta ),\hfill & \hfill in[0,\Theta ],\\ N(k,0)={\int }_0^{\Theta }\zeta (k,\theta )N(k,\theta )d\theta ,\hfill & \hfill in[0,T].\end{array}\right.$$

(2)

A new uncertain stochastic model (2) for APDS is obtained, and we call it hybrid APDS.

The main innovation of this paper is that we first consider a hybrid APDS under chance space. Particularly, we complete the proofs of the existence, uniqueness and stability for a hybrid APDS (2). The proofs of existence and uniqueness are similar to the Refs. [3,26], and the proof of exponential stability is similar to the Refs. [3,27].

In Section 2, we recall some results about Hilbert space, and some concepts, theorems and lemmas about chance space and filtered chance space, which are essential for our analysis. In Section 3, we first give the integral equation of hybrid age-dependent population Equation (2), then prove existence, uniqueness and exponential stability of a strong solution for hybrid APDS (2), respectively.

2. Preliminaries

We will use the variational method, see the Refs. [3,5,26]. Let

$$Z=H^1([0,\Theta ])\equiv \{\psi |\psi \in L^2([0,\Theta ]),\frac{\partial \psi }{\partial z_i}\in L^2([0,\Theta ]),$$

where $\frac{\partial \psi }{\partial z_i}$ is generalized partial derivatives. Z is a Sobolev space. $H=L^2([0,\Theta ])$ satisfies

$$Z\hookrightarrow H\equiv H^{\prime }\hookrightarrow Z^{\prime },$$

where $Z^{\prime }$ is the dual space of Z. $\parallel ·\parallel$, $|·|$ and ${\parallel ·\parallel }_{*}$ stands for the norms in Z, H and $Z^{\prime }$ respectively; $\langle ·,·\rangle$ stands for the duality product between Z, $Z^{\prime }$, and $(·,·)$ stands for the scalar product in H. The constant c satisfies the following inequality:

$$\begin{array}{c}\hfill |z|\le c\parallel z\parallel ,\forall z\in Z.\end{array}$$

(3)

Let $\mathfrak{L}$$(S,H)$ be the space of all bounded linear operators from S into H. For $X\in \mathfrak{L}$$(S,H)$, we use ${\parallel X\parallel }_2$ to represent the Hilbert-Schmidt norm. G denotes a symmetric positive definite matrix, that is,

$${\parallel X\parallel }_2^2=tr(XG{X}^T),$$

Let $\Gamma$ be a nonempty set, and $\mathcal{L}$ a $\sigma$-algebra over $\Gamma$. Each element $\Lambda$ in $\mathcal{L}$ is called an event and $\mathcal{M}\{\Lambda \}$ is the belief degree. Uncertain measure [10] dealing with belief degree satisfies the following axioms:

• Axiom 1. (Normality Axiom) $\mathcal{M}\{\Lambda \}=1$ for the universal set $\Gamma$;
• Axiom 2. (Duality Axiom) $\mathcal{M}\{\Lambda \}+\mathcal{M}\left\{{\Lambda }^c\right\}=1$ for any event $\Lambda$;
• Axiom 3. (Subadditivity Axiom) For every countable sequence of events ${\Lambda }_1,{\Lambda }_2,\cdots$,

  $$\mathcal{M}\left\{\bigcup _{i=1}^{\infty }{\Lambda }_i\right\}\le \sum _{i=1}^{\infty }{\mathcal{M}}_k\left\{{\Lambda }_i\right\}$$

  holds.
• Axiom 4. (Product Axiom) Let $({\Gamma }_j,{\mathcal{L}}_j,{\mathcal{M}}_j)$ be uncertainty spaces for $j=1,2,\cdots$. The product uncertain measure $\mathcal{M}$ is an uncertain measure satisfying

  $$\mathcal{M}\left\{\prod _{j=1}^{\infty }{\Lambda }_j\right\}=\underset{j=1}{\overset{\infty }{\bigwedge }}{\mathcal{M}}_j\left\{{\Lambda }_j\right\}.$$

  where ${\Lambda }_j$ are arbitrary events chosen from ${\mathcal{L}}_j$ for $j=1,2,\cdots$, respectively.

Remark 1.

Axioms 1 and 2 are similar to probability theory while axioms 3 and 4 are fundamentally different from probability theory. In particular, axiom 3 embodies subadditivity, which is different from the additivity of probability theory, and the product axiom of axiom 4 embodies the minimization operation, which is different from the product axiom of probability theory. The detailed analysis can be found in Ref. [10].

Definition 1

([10]). An uncertain variable is a measurable function ξ from an uncertainty space $(\Gamma ,\mathcal{L},\mathcal{M})$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set

$$\{\xi \in B\}=\{\gamma \in \Gamma |\xi (\gamma )\in B\}$$

is an event.

Definition 2

([10]). Let T be an index set and $(\Gamma ,\mathcal{L},\mathcal{M})$ an uncertainty space. An uncertain process is a measurable function from $T\times (\Gamma ,\mathcal{L},\mathcal{M})$ to the set of real numbers such that $\{X_k\in B\}$ is an event for any Borel set B for each time k.

Let $(\Omega ,\mathcal{F},P)$ be a complete probability space with a filtration ${\left\{{\mathcal{F}}_k\right\}}_{k\in [0,T]}$ satisfying the usual conditions, that is, it is increasing and right continuous while ${\mathcal{F}}_0$ contains all P-null sets.

Let $(\Gamma ,\mathcal{L},\mathcal{M})$ be an uncertainty space [10] where normality, duality, subadditivity and product measure axioms are given. Let $C_k$ be a Liu process defined on $(\Gamma ,\mathcal{L},\mathcal{M})$. The process filtration ${\left\{{\mathcal{L}}_k\right\}}_{k\in [0,T]}$ is the sub-$\sigma$-field family (${\mathcal{L}}_k,k\in [0,T]$) of $\mathcal{L}$ satisfying the usual conditions. It is generalized by $\sigma (C_s:s\le k)$ and $\mathcal{M}$-null sets of $\mathcal{L}$, ${\mathcal{L}}_T=\mathcal{L}$.

Liu [20] first introduced chance theory to investigate hybrid systems with uncertainty with respect to belief degree and randomness. To investigate the uncertain stochastic differential systems, Fei [22] extended a filtered chance space $\begin{array}{c}\hfill (\Gamma \times \Omega ,\mathcal{L}\otimes \mathcal{F},{({\mathcal{L}}_k\otimes {\mathcal{F}}_k)}_{k\in [0,T]},\mathcal{M}\times P)\end{array}$ on which some concepts, theorems and lemmas are given as follows.

Definition 3

([22]). (a) Let B be a Borel set, an uncertain random variable ξ is a measurable function from a chance space

$$\begin{array}{c}\hfill (\Gamma \times \Omega ,\mathcal{L}\otimes \mathcal{F},\mathcal{M}\times P)\end{array}$$

to ${\mathbb{R}}^p$ (or ${\mathbb{R}}^{p\times m}$, or H), that is, $\forall B\in {\mathbb{R}}^p$ (or ${\mathbb{R}}^{p\times m}$,or H), the set

$$\begin{array}{c}\hfill \{\xi \in B\}=\{(\gamma ,\omega )\in \Gamma \times \Omega :\xi (\gamma ,\omega )\in B\}\in \mathcal{L}\otimes \mathcal{F}.\end{array}$$

(b) The definition of the expected value $E[\xi ]$ is

$$\begin{array}{cc}\hfill E[\xi ]& =E_P[E_{\mathcal{M}}[\xi ]]\hfill \\ & ={\int }_{\Omega }\left[{\int }_0^{+\infty }\mathcal{M}\{\xi \ge r\}\mathrm{d}r\right]P(d\omega )-{\int }_{\Omega }\left[{\int }_{-\infty }^0\mathcal{M}\{\xi \le r\}\mathrm{d}r\right]P(d\omega )\hfill \end{array}$$

where $E_P$ and $E_{\mathcal{M}}$ denote the expected values under the probability space and the uncertainty space, respectively.

(c) $\forall B$, $\{\xi \in B\}$ is an uncertain random event with chance measure

$$\begin{array}{c}\hfill \mathrm{Ch}\{\xi \in B\}={\int }_0^{1}P\left\{\omega \in \Omega |\mathcal{M}\{\gamma \in \Gamma |\xi (\gamma ,\omega )\in B\}\ge x\right\}\mathrm{d}x\end{array}.$$

Clearly, for constant $\alpha$ and $\beta$, we have $E[\alpha W_k+\beta C_k]=0,$ where $W_k$ is a one-dimensional Wiener process and $C_k$ is a scalar process.

Proposition 1

([28]). (Linearity of Expected Value Operator) Assume ${\eta }_1$ and ${\eta }_2$ are (not necessarily independent) random variables, ${\tau }_1$ and ${\tau }_2$ are independent uncertain variables, and $f_1$ and $f_2$ are measurable functions. Then,

$$\begin{array}{c}\hfill E[f_1({\eta }_1,{\tau }_1)+f_2({\eta }_2,{\tau }_2)]=E[f_1({\eta }_1,{\tau }_1)]+E[f_2({\eta }_2,{\tau }_2)].\end{array}$$

Definition 4.

Assume η is a random variable, τ is an uncertain variable with the regular uncertainty distribution Ψ and a measurable function f, suppose $f(r)$ is strictly increasing as $r\ge 0$, $f(r)$ is strictly decreasing as $r\le 0$, and denote ${\phi }_1$ as the inverse function of $f(r)$ as $r\ge 0$ and ${\phi }_2$ as the inverse function of $f(r)$ as $r\le 0$. The generalized expected value of $f(\eta ,\tau )$ is defined as

$$\begin{array}{c}\hfill E_{CH}[f(\eta ,\tau )]={\int }_{\Omega }\left[{\int }_0^{+\infty }[1-\Psi ({\phi }_1(r))]\mathrm{d}r\right]P(d\omega )+{\int }_{\Omega }\left[{\int }_0^{+\infty }[\Psi ({\phi }_2(r))]\mathrm{d}r\right]P(d\omega )\end{array}$$

Remark 2.

If $f_i({\eta }_i,{\tau }_i)\in C(\mathbb{R},\mathbb{R},{\mathbb{R}}^{+})$ satisfies $f_i(0)=0$, then we can easily obtain $E_{CH}({\displaystyle \sum _{i=1}^p}{f}_i({\eta }_i,{\tau }_i))\le p{\displaystyle \sum _{i=1}^p}{E}_{CH}[f_i({\eta }_i,{\tau }_i)]$ by the subadditivity of the uncertain random measure.

Definition 5

([22]). (a) For each time $k\in [0,T],$ if $X_k$ is an uncertain random variable, then we call $X_k$ an uncertain stochastic process. If the sample paths of $X_k$ are continuous functions of k for almost all $(\gamma ,\omega )\in \Gamma \times \Omega ,$ then we call it continuous.

(b) If $X(k,\gamma )$ is ${\mathcal{F}}_k$-measurable for all $k\in [0,T],$$\gamma \in \Gamma ,$ then we call it ${\mathcal{F}}_k$-adapted. Further, if $X(k)$ is ${\mathcal{L}}_k\otimes {\mathcal{F}}_k$-measurable for all $k\in [0,T],$ then we call it ${\mathcal{L}}_k\otimes {\mathcal{F}}_k$-adapted (or adapted).

(c) If the uncertain stochastic process is measurable related to the σ-algebra

$$\begin{array}{cc}\hfill & \Im ({\mathcal{L}}_k\otimes {\mathcal{F}}_k)\hfill \\ \hfill =& \{A\in B([0,T])\otimes \mathcal{L}\otimes \mathcal{F}:A\cap ([0,k]\times \Gamma \times \Omega )\in B([0,k])\otimes {\mathcal{L}}_k\otimes {\mathcal{F}}_k\},\hfill \end{array}$$

then we call it progressively measurable.

Further, if the uncertain stochastic process $X(k):\Gamma \times \Omega \to {\mathbb{R}}^p$ (or $X(k):\Gamma \times \Omega \to {\mathbb{R}}^{p\times m}$ is progressively measurable and satisfies $\forall k\in [0,T],$$E_{CH}[{\int }_0^T|X_k{|}^{2}dk]<\infty$, then we call it $L^2$-progressively measurable, where $L^2(0,T;{\mathbb{R}}^p)$ (or $L^2(0,T;{\mathbb{R}}^{p\times m}$)) denotes the set of $L^2$-progressively measurable uncertain random processes. The extension of ${\mathbb{R}}^p$ or ${\mathbb{R}}^{p\times m}$ to a Hilbert space can be omitted here. Later, we will give the definition in the next section.

Next, we redefine the Liu integral in the mean square sense according to Remark 2 of Ref. [29].

Definition 6

([29]). Let $X_k$ be an uncertain process and $C_k$ be a Liu process. For any partition of closed interval $[a,b]$ with $a=k_1<k_2<\cdots <k_{n+1}=b$, the mesh is written as

$$\Delta =\underset{1\le i\le n}{max}|k_{i+1}-k_i|.$$

Then uncertain integral of $X_k$ with respect to $C_k$ is

$${\int }_a^b{X}_{k}d{C}_k=\underset{\Delta \to 0}{lim}\sum _{i=1}^n{X}_{k_i}·(C_{k_{i+1}}-C_{k_i}),$$

provided that the limit exists in the mean square sense and is finite. In this case, the uncertain process $X_k$ is said to be integrable.

Definition 7

([21]). Let $W_k$ be a Wiener process and $C_k$ a Liu process. Then, ${\mathcal{H}}_k=(W_k,C_k)$ is called a Wiener-Liu process. The Wiener-Liu process is said to be standard if both $W_k$ and $C_k$ are standard.

Definition 8

([21]). Let $X_k=({\widehat{X}}_k,{\tilde{X}}_k)$ where ${\widehat{X}}_k$ and $\tilde{X}$ are scalar uncertain stochastic processes, and let ${\mathcal{H}}_k=(W_k,C_k)$ be a standard Wiener-Liu process. For any partition of closed interval $[a,b]$ with $a=k_1<k_2<\cdots <k_{N+1}=b$, the mesh is written as

$$\begin{array}{c}\hfill \Delta =\underset{1\le i\le N}{max}|k_{i+1}-k_i|.\end{array}$$

Then uncertain stochastic integral of $X_k$ with respect to $H_k$ is

$$\begin{array}{cc}\hfill {\int }_a^b{X}_{k}d{\mathcal{H}}_k=& \underset{\Delta \to 0}{lim}\sum _{i=1}^N({\widehat{X}}_{k_i}·(W_{k_{i+1}}-W_{k_i})+{\tilde{X}}_{k_i}·(C_{k_{i+1}}-C_{k_i})),\hfill \end{array}$$

provided that the limit exists in mean square and is a uncertain random variable.

Remark 3.

The uncertain stochastic integral may also be written as follows:

$$\begin{array}{c}\hfill {\int }_a^b{X}_{k}d{\mathcal{H}}_k={\int }_a^b({\widehat{X}}_{k}d{W}_k+{\tilde{X}}_{k}d{C}_k).\end{array}$$

(4)

The following theorem will give the Itô-Liu formula of the 1-dimensional case.

Proposition 2

([21]). Let ${\mathcal{H}}_k$ be a Wiener-Liu process given by

$$\begin{array}{c}\hfill {\mathcal{H}}_k=(X_k,Y_k)=({\mu }_{1}k+{\sigma }_1{W}_k,{\mu }_{2}k+{\sigma }_2{C}_k).\end{array}$$

Let $W_k$ be a Wiener process and $C_k$ a Liu process and $g(k,x,y)$ a twice continuously differentiable function. Define $G_k=g(k,X_k,Y_k).$ Then, we have the following chain rule:

$$\begin{array}{cc}\hfill d{G}_k=& \frac{\partial g}{\partial k}(k,X_k,Y_k)dk+\frac{\partial g}{\partial x}(k,X_k,Y_k)d{W}_k+\frac{\partial g}{\partial y}(k,X_k,Y_k)d{C}_k\hfill \\ & +\frac{1}{2}\frac{{\partial }^{2}g}{\partial x^2}(k,X_k,Y_k)dk.\hfill \end{array}$$

Corollary 1

([21]). The infinitesimal increments $d{W}_k$ and $d{C}_k$ in (28) may be replaced with the derived Wiener-Liu process,

$$\begin{array}{c}\hfill X_k={\int }_0^k{\mu }_{u}du+{\int }_0^k{\alpha }_{u}d{W}_u+{\int }_0^k{\beta }_{u}d{C}_u\end{array}$$

where ${\mu }_k$ and ${\beta }_k$ are absolutely integrable uncertain stochastic processes, and ${\alpha }_k$ is a square integrable uncertain stochastic process, then $\forall \Phi \in {\mathcal{C}}^2(\mathbb{R})$, (${\mathcal{C}}^2$ means second order continuous differentiable), thus producing

$$\begin{array}{cc}\hfill \Phi (X_k)=& \Phi (X_0)+{\int }_0^k{\Phi }^{\prime }(X_u){\mu }_{u}du+{\int }_0^k{\Phi }^{\prime }(X_u){\alpha }_{u}d{W}_u\hfill \\ & +{\int }_0^k{\Phi }^{\prime }(X_u){\beta }_{u}d{C}_u+\frac{1}{2}{\int }_0^k{\Phi }^{″}(X_u){\alpha }_u^{2}du.\hfill \end{array}$$

Let $W_k=(W_{1k},W_{2k},\cdots ,W_{pk})$ be a p-dimensional standard Wiener process, and $C_k=(C_{1k},C_{2k},\cdots ,C_{qk})$ a q-dimensional standard Liu process. If $r_i$ and $z_{ij}$ are absolute integrable hybrid processes, and $w_{ij}$ are square integrable hybrid processes, for $i=1,2,\cdots ,m,$ $j=1,2,\cdots ,q,$ then the m-dimensional hybrid process ${\mathbf{X}}_k=(X_{1k},X_{2k},\cdots ,X_{mk})$ is given by

$$\left\{\begin{array}{c}d{X}_{1k}=r_{1}dk+{\displaystyle \sum _{j=1}^p}{w}_{1j}d{W}_{jk}+{\displaystyle \sum _{j=1}^q}{z}_{1j}d{C}_{jk}\hfill \\ ⋮⋮⋮\hfill \\ d{X}_{mk}=r_{m}dk+{\displaystyle \sum _{j=1}^p}{w}_{mj}d{W}_{jk}+{\displaystyle \sum _{j=1}^q}{z}_{mj}d{C}_{jk}.\hfill \end{array}\right.$$

Or, in matrix notation, simply,

$$d{\mathbf{X}}_k=\mathbf{r}dk+\mathbf{w}d{\mathbf{W}}_k+\mathbf{z}d{\mathbf{C}}_k,$$

where

$$\mathbf{r}=\left(\begin{array}{c}{r}_1\\ ⋮\\ r_m\end{array}\right),\mathbf{w}=\left(\begin{array}{c}{w}_{11}\cdots w_{1p}\\ ⋮⋮\\ w_{m1}\cdots w_{mp}\end{array}\right),\mathbf{z}=\left(\begin{array}{c}{z}_{11}\cdots z_{1q}\\ ⋮⋮\\ z_{m1}\cdots z_{mq}\end{array}\right),d{\mathbf{W}}_k=\left(\begin{array}{c}d{W}_{1k}\\ ⋮\\ d{W}_{pk}\end{array}\right),d{\mathbf{C}}_k=\left(\begin{array}{c}d{C}_{1k}\\ ⋮\\ d{C}_{qk}\end{array}\right).$$

Proposition 3

([21]). Assume m-dimensional hybrid process ${\mathbf{X}}_k$ is given by

$$d{\mathbf{X}}_k=\mathbf{r}dk+\mathbf{w}d{\mathbf{W}}_k+\mathbf{z}d{\mathbf{C}}_k.$$

Let $g(k,x_1,\cdots ,x_m)$ be a multivariate continuously differentiable function. Define $G_k=g(k,X_{1k},\cdots ,X_{mk})$. Then,

$$\begin{array}{cc}\hfill d{G}_k=& \frac{\partial g}{\partial k}(k,X_{1k},\cdots ,X_{mk})dk+\sum _{i=1}^m\frac{\partial g}{\partial x_i}(k,X_{1k},\cdots ,X_{mk})d{X}_{ik}\hfill \\ & +\frac{1}{2}\sum _{i=1}^m\sum _{j=1}^m\frac{{\partial }^{2}g}{\partial x_i\partial x_j}(k,X_{1k},\cdots ,X_{mk})d{X}_{ik}d{X}_{jk},\hfill \end{array}$$

where $d{W}_{ik}d{W}_{jk}=I_{ij}dk$, $d{W}_{ik}dk=dkd{W}_{ik}=d{C}_{ık}d{C}_{jk}=dkd{C}_{ık}=d{W}_{ik}d{C}_{ık}=0$, for $i,j=1,2,\cdots ,p,$ $𝚤,𝚥=1,2,\cdots ,q.$ Here,

$$I_{ij}=\left\{\begin{array}{c}0,i\ne j\hfill \\ 1,i=j\hfill \end{array}\right.$$

In other words, it can be expressed as

$$\begin{array}{cc}\hfill d{G}_k=& \frac{\partial g}{\partial k}(k,X_{1k},\cdots ,X_{mk})dk+\sum _{i=1}^p\frac{\partial g}{\partial x_i}(k,W_{1k},\cdots ,W_{pk},C_{1k},\cdots ,C_{qk})d{W}_{ik}\hfill \\ & +\sum _{j=1}^q\frac{\partial g}{\partial x_{m+j}}(k,W_{1k},\cdots ,W_{pk},C_{1k},\cdots ,C_{qk})d{C}_{jk}+\frac{1}{2}\sum _{i=1}^p\frac{{\partial }^{2}g}{\partial x_i^2}(k,W_{1k},\cdots ,W_{pk},C_{1k},\cdots ,C_{qk})dk,\hfill \end{array}$$

By Pardoux [30] and Ichikawa [31], the extension of the general Itô-Liu formula can be omitted here.

Lemma 1

([12] (Liu’s lemma)). Let $C_k$ be a Liu process, and $X_k$ be an integrable uncertain process on $[a,b]$ with respect to time k. Then, the inequality

$$|{\int }_a^b{X}_{k}d{C}_k|\le \mathcal{K}{\int }_a^b|X_k|dk$$

holds, where $\mathcal{K}$ is the nonnegative Lipschitz constant of the sample path $C_k$.

Corollary 2

(the extension of Liu’s lemma). Let $C_k$ be a Liu process, and $X_k$ be an integrable uncertain stochastic process on $[t_0,t]$ with respect to k, and $t\le T.$ Then, the inequality

$$E_{\mathcal{M}}[\underset{t_0\le t\le T}{sup}|{\int }_{t_0}^t{X}_{k}d{C}_k|]\le L\sqrt{E_{\mathcal{M}}\left[{\left|{\int }_{t_0}^T|X_k|dk\right|}^2\right]}$$

holds, where L is a nonnegative constant.

Proof. 

Similar to the proof of Lemma 1, we can obtain

$$|{\int }_{t_0}^t{X}_{k}d{C}_k|\le \mathcal{K}{\int }_{t_0}^t|X_k|dk,$$

where $\mathcal{K}$ is the nonnegative Lipschitz constant of the sample path $C_k$, then $\mathcal{K}$ is an uncertain variable on $(\Gamma ,\mathcal{L},\mathcal{M})$; thus,

$$\underset{t_0\le t\le T}{sup}|{\int }_{t_0}^t{X}_{k}d{C}_k|\le \mathcal{K}\underset{t_0\le t\le T}{sup}{\int }_{t_0}^t|X_k|dk.$$

So, taking the expected values on both sides, we obtain

$$E_{\mathcal{M}}[\underset{t_0\le t\le T}{sup}|{\int }_{t_0}^t{X}_{k}d{C}_k|]\le E_{\mathcal{M}}[\mathcal{K}\underset{t_0\le t\le T}{sup}{\int }_{t_0}^t|X_k|dk]\le E_{\mathcal{M}}[\mathcal{K}{\int }_{t_0}^T|X_k|dk].$$

In addition, by Holder’s inequality [10],

$$\begin{array}{c}\hfill E_{\mathcal{M}}[\mathcal{K}{\int }_{t_0}^T|X_k|dk]\le \sqrt{E_{\mathcal{M}}[{\mathcal{K}}^2]}\sqrt{E_{\mathcal{M}}\left[{\left|{\int }_{t_0}^T|X_k|dk\right|}^2\right]}.\end{array}$$

By using Theorem 2 of Ref. [13], we know that

$$\begin{array}{cc}\hfill \mathcal{M}\{\mathcal{K}\ge x\}& =1-\mathcal{M}\{\mathcal{K}\le x\}\hfill \\ \hfill & \le 1-(2{(1+exp(\frac{-\pi x}{\sqrt{3}}))}^{-1}-1)\hfill \\ \hfill & =1-\frac{2}{1+exp(\frac{-\pi x}{\sqrt{3}})}+1\hfill \\ \hfill & =2(1-\frac{1}{1+exp(\frac{-\pi x}{\sqrt{3}})})\hfill \\ \hfill & =2(\frac{exp(\frac{-\pi x}{\sqrt{3}})}{1+exp(\frac{-\pi x}{\sqrt{3}})})\hfill \\ \hfill & =2{(1+exp(\frac{\pi x}{\sqrt{3}}))}^{-1}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill E_{\mathcal{M}}[{\mathcal{K}}^2]& ={\int }_0^{+\infty }\mathcal{M}\{{\mathcal{K}}^2\ge x\}dx\hfill \\ \hfill & ={\int }_0^{+\infty }\mathcal{M}\{\mathcal{K}\ge \sqrt{x}\}dx\hfill \\ \hfill & \le 2{\int }_0^{+\infty }\frac{1}{1+exp(\frac{\pi \sqrt{3x}}{3})}dx(\mathrm{let}\frac{\pi \sqrt{3x}}{3}=u),\hfill \\ \hfill & ={\int }_0^{+\infty }\frac{3}{{\pi }^2}\frac{u}{1+e^u}du,\hfill \end{array}$$

while

$$\underset{u\to +\infty }{lim}\frac{u}{1+e^u}/\frac{1}{(u^2+1)}=\underset{u\to +\infty }{lim}\frac{u(u^2+1)}{1+e^u}=0.$$

Since ${\int }_0^{+\infty }\frac{1}{u^2+1}du$ is convergent, so ${\int }_0^{+\infty }\frac{u}{1+e^u}du$ is convergent. Consequently, it holds that

$$E_{\mathcal{M}}[{\mathcal{K}}^2]<+\infty ,$$

where, we simply denote $E_{\mathcal{M}}[{\mathcal{K}}^2]$ by L. So, it holds that

$$E_{\mathcal{M}}[\underset{t_0\le t\le T}{sup}|{\int }_{t_0}^t{X}_{k}d{C}_k|]\le L\sqrt{E_{\mathcal{M}}\left[{\left|{\int }_{t_0}^T|X_k|dk\right|}^2\right]}.$$

The proof of the corollary is completed. □

Lemma 2

([21]). By using the Itô-Liu uncertain stochastic integral and the expectations operator $E_{CH}[·]$, then, $\forall {\tau }_1,{\tau }_2\in [0,T]$, $X\in L^2(0,T;{\mathbb{R}}^{p\times d})$, and $Y\in L^2(0,T;{\mathbb{R}}^{p\times m})$,

$$\begin{array}{c}\hfill E_{CH}[{\int }_{{\tau }_1}^{{\tau }_2}{X}_{k}d{C}_k]=0,E_{CH}[{\int }_{{\tau }_1}^{{\tau }_2}{Y}_{k}d{W}_k]=0,\end{array}$$

where ${\mathbb{R}}^{p\times d},{\mathbb{R}}^{p\times m}$ space can extend to Hilbert space, see [32] (omitted here). We will use these results in the next section.

Lemma 3

(Gronwall inequality [33]). Let $\mu (k)$ be a nonnegative function such as

$$\mu (k)\le A+D{\int }_0^k\mu (r)dr,\forall k\in [0,T]$$

for some constants $A,D$. Then, we have

$$\mu (k)\le Aexp(Dk),\forall k\in [0,T].$$

3. Main Results

In this paper, let $\mathcal{C}=\mathcal{C}([0,T],H)$ be the space of all continuous functions from $[0,T]$ into H with sup-norm ${\parallel \phi \parallel }_{\mathcal{C}}=\underset{0\le r\le T}{sup}|\phi |(r)$, $L_Z^p=L^p([0,T];Z)$ and $L_H^p=L^p([0,T];H)$.

Here, we give the equivalent uncertain stochastic integral Equation (5) of (2):

$$\left\{\begin{array}{cc}\hfill N_k=& N_0-{\int }_0^k\frac{\partial N_r}{\partial \theta }dr-{\int }_0^k\eta (r,\theta )N_{r}dr+{\int }_0^{k}f(r,N_r)dr+{\int }_0^{k}g(r,N_r)d{W}_r\hfill \\ & +{\int }_0^{k}h(r,N_r)d{C}_r,\forall k\in [0,T],\hfill \\ \hfill N(k,0)=& {\int }_0^{\Theta }\zeta (k,\theta )N_{r}d\theta ,\forall k\in [0,T].\hfill \end{array}\right.$$

(5)

where $N_r=N(r,\theta )$, $N_0=N(0,\theta )$.

Definition 9.

Suppose that $N_0$ is an uncertain random variable such that $E|N_0{|}^2<\infty$. An uncertain stochastic process $N_k$ is said to be a strong solution on $\Gamma \times \Omega$ to the uncertain stochastic integral Equation (5) for $k\in [0,T]$ if the following $(a),(b),(c)$ hold:

(a) $N_k$ is a ${\mathcal{L}}_k\otimes {\mathcal{F}}_k$-measurable uncertain random variable $a.e.$ in time k (we denote by $a.e.k$ in the sequel);

(b) $N_k\in I^2(0,T;Z)\bigcap L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))$, $T>0$, where $I^2(0,T;Z)$ stands for the space of all Z-valued processes ${(N_k)}_{k\in [0,T]}$ ($N_k$ for short) measurable (from $[0,T]\times \Gamma \times \Omega$ into Z), and satisfying

$$E_{CH}{\int }_0^T{\parallel N_k\parallel }^{2}dk<\infty .$$

Here, $\mathcal{C}(0,T;H)$ stands for the space of all continuous functions from $[0,T]$ to H;

(c) $\forall k\in [0,T]$, Equation (5) is satisfied with chance measure one.

If T is replaced by ∞, $N_k$ is called a global strong solution of (5).

Let Q denote $I^2(0,T;V)\bigcap L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))$. We will next prove that there exists a unique $N_k\in Q$ such that (5) holds. Firstly, we give some assumptions:

Let $f(k,·),h(k,·):L_H^2\to H$ be a family of nonlinear operators defined a.e.k., and they satisfy (c.1) and Lipschitz condition (c.2):

(c.1) $f(k,0)=0,h(k,0)=0$;

(c.2) $\exists {\lambda }_1>0$ such that $x,y\in \mathcal{C},$

$$|f(k,y)-f(k,x)|\vee |h(k,y)-h(k,x)|\le {\lambda }_1{\parallel y-x\parallel }_{\mathcal{C}},a.e.k.$$

Let $g(k,·):L_H^2\to$$\mathfrak{L}$$(S,H)$ be a family of nonlinear operators defined a.e.k., $g(k,x)\in \mathfrak{L}(S,H)$, and they satisfy (c.3) and Lipschitz condition (c.4)

(c.3) $g(k,0)=0$

(c.4) $\exists {\lambda }_2>0$ such that $x,y\in \mathcal{C}$

${\parallel g(k,y)-g(k,x)\parallel }_2\le {\lambda }_2{\parallel y-x\parallel }_{\mathcal{C}},a.e.k$,

(c.5) $\eta (k,\theta )$, $\zeta (k,\theta )$ are nonnegative measurable, and

$$\left\{\begin{array}{cc}0\le {\eta }_0\le \eta (k,\theta )<\infty \hfill & inU,\hfill \\ 0\le \zeta (k,\theta )\le \overline{\zeta }<\infty \hfill & inU.\hfill \end{array}\right.$$

(c.6) There exist constants $\varsigma >0$, $\rho >0$, $\lambda \in \Re$, and a nonnegative continuous function $\chi (k)$, $k\in R^{+}$, such that

$$2\langle f(k,z),z\rangle +{\parallel g(k,z)\parallel }_2^2\le -{\varsigma \parallel z\parallel }^2+\lambda {|z|}^2+\chi (k)e^{-\rho k},z\in Z,a.e.k..$$

Here, $\forall ϵ>0$, $\chi (k)$ satisfies $\chi (k)=o(e^{ϵk})$, as $k\to \infty ,$ i.e., $\underset{k\to \infty }{lim}\chi (k)/e^{ϵk}=0$.

3.1. Uniqueness Theorem

Theorem 1.

(Uniqueness) Assume (c.1)–(c.5) hold. Then, there exists a unique solution of (5) in Q.

Proof. 

Suppose that $N_{1k},N_{2k}\in Q$ are two solutions of (5). Then, for $|N_{1k}-N_{2k}{|}^2$, it holds from Itô-Liu’s formula that

$$\begin{array}{cc}\hfill & |N_{1k}-N_{2k}{|}^2\hfill \\ \hfill =& 2{\int }_0^k\langle -\frac{\partial N_{1r}}{\partial \theta }+\frac{\partial N_{2r}}{\partial \theta }-\eta (r,\theta )(N_{1r}-N_{2r}),N_{1r}-N_{2r}\rangle dr\hfill \\ \hfill & +2{\int }_0^k(f(r,N_{1r})-f(r,N_{2r}),N_{1r}-N_{2r})dr+{\int }_0^k{\parallel g(r,N_{1r})-g(r,N_{2r})\parallel }_2^{2}dr\hfill \\ \hfill & +2{\int }_0^k(N_{1r}-N_{2r},(g(r,N_{1r})-g(r,N_{2r}))d{W}_r)\hfill \\ & +2{\int }_0^k(N_{1r}-N_{2r},(h(r,N_{1r})-h(r,N_{2r}))d{C}_r)\hfill \\ \hfill \le & -2{\int }_0^k\langle \frac{\partial (N_{1r}-N_{2r})}{\partial \theta },N_{1r}-N_{2r}\rangle dr-2{\eta }_0{\int }_0^k(N_{1r}-N_{2r},N_{1r}-N_{2r})dr\hfill \\ \hfill & +2{\int }_0^k(f(r,N_{1r})-f(r,N_{2r}),N_{1r}-N_{2r})dr+{\int }_0^k{\parallel g(r,N_{1r})-g(r,N_{2r})\parallel }_2^{2}dr\hfill \\ \hfill & +2{\int }_0^k(N_{1r}-N_{2r},(g(r,N_{1r})-g(r,N_{2r}))d{W}_r)\hfill \\ & +2{\int }_0^k(N_{1r}-N_{2r},(h(r,N_{1r})-h(r,N_{2r}))d{C}_r)\hfill \end{array}$$

because

$$\begin{array}{cc}\hfill & -\langle \frac{\partial (N_{1r}-N_{2r})}{\partial \theta },N_{1r}-N_{2r}\rangle \hfill \\ \hfill & =-{\int }_0^{\Theta }\frac{\partial (N_{1r}-N_{2r})}{\partial \theta }(N_{1r}-N_{2r})d\theta \hfill \\ \hfill & =-{\int }_0^{\Theta }(N_{1r}-N_{2r})d_{\theta }(N_{1r}-N_{2r})\hfill \\ \hfill & =-\frac{1}{2}{(N_{1r}-N_{2r})}^2{|}_0^{\Theta }\hfill \\ \hfill & =0-(-\frac{1}{2}{({\int }_0^{\Theta }\zeta (k,\theta )(N_{1r}-N_{2r})d\theta )}^2)\hfill \\ \hfill & =\frac{1}{2}{({\int }_0^{\Theta }\zeta (k,\theta )(N_{1r}-N_{2r})d\theta )}^2\hfill \\ \hfill & \le \frac{1}{2}{\int }_0^{\Theta }{\zeta }^2(r,\theta )d\theta {\int }_0^{\Theta }{(N_{1r}-N_{2r})}^{2}d\theta \hfill \\ \hfill & \le \frac{1}{2}\Theta {\overline{\zeta }}^2{|N_{1r}-N_{2r}|}^2.\hfill \end{array}$$

Thus, we obtain that

$$\begin{array}{cc}\hfill & |N_{1k}-N_{2k}{|}^2\hfill \\ \hfill & \le \Theta {\overline{\zeta }}^2{\int }_0^k|N_{1r}-N_{2r}{|}^{2}dr+2{\int }_0^k|N_{1r}-N_{2r}||f(r,N_{1r})-f(r,N_{2r})|dr\hfill \\ \hfill & -2{\eta }_0{\int }_0^k|N_{1r}-N_{2r}{|}^{2}dr+{\int }_0^k{\parallel g(r,N_{1r})-g(r,N_{2r})\parallel }_2^{2}dr\hfill \\ \hfill & +2{\int }_0^k(N_{1r}-N_{2r},(g(r,N_{1r})-g(r,N_{2r})d{W}_r)+2{\int }_0^k(N_{1r}-N_{2r},(h(r,N_{1r})-h(r,N_{2r})d{C}_r).\hfill \end{array}$$

(6)

Then, $\forall k\in [0,T]$, by (c.2) and (c.4), it holds that

$$\begin{array}{cc}\hfill & E_{CH}\underset{0\le r\le k}{sup}{|N_{1r}-N_{2r}|}^2\hfill \\ \hfill & \le 4((|\Theta {\overline{\zeta }}^2-2{\eta }_0|+1){\int }_0^k{E}_{CH}|N_{1r}-N_{2r}{|}^{2}dr+({\lambda }_1^2+{\lambda }_2^2){\int }_0^k{E}_{CH}{\parallel N_{1r}-N_{2r}\parallel }_C^{2}dr\hfill \\ \hfill & +2E_{CH}\underset{0\le r\le k}{sup}{\int }_0^r(N_{1s}-N_{2s},(g(s,N_{1s})-g(s,N_{2s}))d{W}_s)\hfill \\ \hfill & +2E_{CH}\underset{0\le r\le k}{sup}{\int }_0^r(N_{1s}-N_{2s},(h(s,N_{1s})-h(s,N_{2s}))d{C}_s)).\hfill \end{array}$$

(7)

By BDG inequality, it holds that

$$\begin{array}{cc}\hfill & E_{CH}[\underset{0\le r\le k}{sup}{\int }_0^r(N_{1s}-N_{2s},(g(s,N_{1s})-g(s,N_{2s}))d{W}_s)]\hfill \\ \hfill & \le 3E_{CH}[\underset{0\le r\le k}{sup}|N_{1r}-N_{2r}|({\int }_0^k\parallel g(r,N_{1r})-g(r,N_{2r}){\parallel }_2^{2}dr{)}^{1/2}]\hfill \\ \hfill & \le \frac{1}{6}{E}_{CH}[\underset{0\le r\le k}{sup}|N_{1r}-N_{2r}{|}^2+K_1{\int }_0^k\parallel g(r,N_{1r})-g(r,N_{2r}){\parallel }_2^2]dr\hfill \\ \hfill & \le \frac{1}{6}{E}_{CH}[\underset{0\le r\le k}{sup}|N_{1r}-N_{2r}{|}^2+K_1·{\lambda }_2^2{\int }_0^k{E}_{CH}{\parallel N_{1r}-N_{2r}\parallel }_C^{2}dr.\hfill \end{array}$$

(8)

By the extension of Liu’s lemma (Corollary 2), it holds that

$$\begin{array}{cc}\hfill & E_{CH}[\underset{0\le r\le k}{sup}{\int }_0^r(N_{1s}-N_{2s},(h(s,N_{1s})-h(s,N_{2s}))d{C}_s)]\hfill \\ \hfill & \le L{E}_{CH}[\underset{0\le r\le k}{sup}|N_{1r}-N_{2r}|({\int }_0^k|h(r,N_{1r})-h(r,N_{2r}){|}^{2}dr{)}^{1/2}]\hfill \\ \hfill & \le \frac{1}{6}{E}_{CH}[\underset{0\le r\le k}{sup}|N_{1r}-N_{2r}{|}^2+K_2{\int }_0^k|h(r,N_{1r})-h(r,N_{2r}){|}^2]dr\hfill \\ \hfill & \le \frac{1}{6}{E}_{CH}[\underset{0\le r\le k}{sup}|N_{1r}-N_{2r}{|}^2+K_2·{\lambda }_1^2{\int }_0^k{E}_{CH}{\parallel N_{1r}-N_{2r}\parallel }_C^{2}dr,\hfill \end{array}$$

(9)

where $K_1,K_2$ are positive constants.

Substituting (8) and (9) into (7) yields

$$\begin{array}{cc}\hfill & E_{CH}\underset{0\le r\le k}{sup}{|N_{1r}-N_{2r}|}^2\hfill \\ & \le 12(|\Theta {\overline{\zeta }}^2-2{\eta }_0|+1+{\lambda }_1^2+{\lambda }_2^2+2K_1{\lambda }_2^2+2K_2{\lambda }_1^2){\int }_0^k{E}_{CH}\underset{0\le s\le r}{sup}{|N_{1s}-N_{2s}|}^{2}dr,\forall k\in [0,T].\hfill \end{array}$$

By Gronwall inequality (Lemma 3), we get the uniqueness. □

3.2. Existence Theorem

We firstly consider the following equations

$$\begin{array}{c}\hfill N_k^{(1)}=N_0+{\int }_0^k[-\frac{\partial N_r^{(1)}}{\partial \theta }-\frac{\Theta {\overline{\zeta }}^2}{2}{N}_r^{(1)}]dr,k\in [0,T],\\ \hfill N^{(1)}(k,0)={\int }_0^{\Theta }\zeta (k,\theta )N_k^{(1)}d\theta ,k\in [0.T],\end{array}$$

(10)

$$\begin{array}{cc}\hfill N_k^{(n+1)}=& N_0+{\int }_0^k[-\frac{\partial N_r^{(n+1)}}{\partial \theta }-\frac{\Theta {\overline{\zeta }}^2}{2}{N}_r^{(n+1)}]dr+{\int }_0^k\frac{\Theta {\overline{\zeta }}^2}{2}{N}_r^{(n)}dr-{\int }_0^k\eta (r,\theta )N_r^{(n)}dr\hfill \\ \hfill & +{\int }_0^{k}f(r,N_r^{(n)})dr+{\int }_0^{k}g(r,N_r^{(n)})d{W}_r+{\int }_0^{k}h(r,N_r^{(n)})d{C}_r,k\in [0,T],\forall n\ge 1,\hfill \end{array}$$

(11)

$$\begin{array}{c}\hfill N^{(n+1)}(k,0)={\int }_0^{\Theta }\zeta (k,\theta )N_k^{(n+1)}d\theta ,k\in [0,T],\forall n\ge 1.\end{array}$$

(12)

Remark 4.

These equations are a sequence of successive approximations of functions. We will prove the existence theorem by a successive approximation method.

Before we prove the existence theorem, we first prove three lemmas.

Lemma 4.

Assume the conditions $(c.1),(c.2),(c.3),(c.4),(c.5)$ hold. Then, $\left\{N_k^{(n)}\right\}$ is a Cauchy sequence in $L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))$.

Proof. 

For $n>1$ and the uncertain stochastic process $N_k^{(n+1)}-N_k^{(n)}$, by Itô-Liu’s formula, it holds that

$$\begin{array}{cc}\hfill & |N_k^{(n+1)}-N_k^{(n)}{|}^2\hfill \\ \hfill =& 2{\int }_0^k\langle -\frac{\partial N_r^{(n+1)}}{\partial \theta }+\frac{\partial N_r^{(n)}}{\partial \theta },N_r^{(n+1)}-N_r^{(n)}\rangle dr-2{\int }_0^k(\eta (r,\theta )(N_r^{(n)}-N_r^{(n-1)}),N_r^{(n+1)}-N_r^{(n)})dr\hfill \\ \hfill & -\Theta {\overline{\zeta }}^2{\int }_0^k{|N_r^{(n+1)}-N_r^{(n)}|}^{2}dr+\Theta {\overline{\zeta }}^2{\int }_0^k(N_r^{(n+1)}-N_r^{(n)},N_r^{(n)}-N_r^{(n-1)})dr\hfill \\ \hfill & +2{\int }_0^k(f(N_r^{(n)})-f(N_r^{(n-1)}),N_r^{(n+1)}-N_r^{(n)})dr+2{\int }_0^k(N_r^{(n+1)}-N_r^{(n)},(g(N_r^{(n)})-g(N_r^{(n-1)}))d{W}_r)\hfill \\ \hfill & +2{\int }_0^k(N_r^{(n+1)}-N_r^{(n)},(h(N_r^{(n)})-h(N_r^{(n-1)}))d{C}_r)+{\int }_0^k{\parallel g(N_r^{(n)})-g(N_r^{(n-1)})\parallel }_2^{2}dr.\hfill \end{array}$$

(13)

where, by definition, $N_k^{(n)}:=N^{(n)}(k,\theta )$, $f(N_k^{(n)}):=f(k,N_k^{(n)})$, $g(N_k^{(n)}):=g(k,N_k^{(n)})$ and $h(N_k^{(n)}):=h(k,N_k^{(n)})$. It is easy to deduce

$$\begin{array}{cc}\hfill & |N_k^{(n+1)}-N_k^{(n)}{|}^2\hfill \\ \hfill \le & |\Theta {\overline{\zeta }}^2-2{\eta }_0|{\int }_0^k|N_r^{(n+1)}-N_r^{(n)}||N_r^{(n)}-N_r^{(n-1)}|dr\hfill \\ & +2|{\int }_0^k(N_r^{(n+1)}-N^{(n)},(g(N_r^{(n)})-g(N_r^{(n-1)}))d{W}_r)|\hfill \\ \hfill & +2|{\int }_0^k(N_r^{(n+1)}-N^{(n)},(h(N_r^{(n)})-h(N_r^{(n-1)}))d{C}_r)|\hfill \\ & +2{\int }_0^k|f(N_r^{(n)})-f(N_r^{(n-1)})||N_r^{(n+1)}-N_r^{(n)}|dr\hfill \\ & +{\int }_0^k{\parallel g(N_r^{(n)})-g(N_r^{(n-1)})\parallel }_2^{2}dr.\hfill \end{array}$$

(14)

Consequently, (14) yields

$$\begin{array}{cc}\hfill & E_{CH}[\underset{0\le a\le k}{sup}|N_a^{(n+1)}-N_a^{(n)}{|}^2]\hfill \\ \hfill \le & 5(|\Theta {\overline{\zeta }}^2-2{\eta }_0|E_{CH}{\int }_0^k|N_r^{(n+1)}-N_r^{(n)}||N_r^{(n)}-N_r^{(n-1)}|dr\hfill \\ & +2E_{CH}[\underset{0\le \theta \le k}{sup}|{\int }_0^{\theta }(N_r^{(n+1)}-N_r^{(n)},(g(N_r^{(n)})-g(N_r^{(n-1)}))d{W}_r)|\hfill \\ \hfill & +2E_{CH}[\underset{0\le \theta \le k}{sup}|{\int }_0^{\theta }(N_r^{(n+1)}-N_r^{(n)},(h(N_r^{(n)})-h(N_r^{(n-1)}))d{C}_r)|\hfill \\ & +2E_{CH}{\int }_0^k|f(N_r^{(n)})-f(N_r^{(n-1)})||N_r^{(n+1)}-N_r^{(n)}|dr\hfill \\ \hfill & +E_{CH}{\int }_0^k\parallel g(N_r^{(n)})-g(N_r^{(n-1)}){\parallel }_2^{2}dr).\hfill \end{array}$$

(15)

Next, we will estimate the five terms of (15) by using the inequality

$$\begin{array}{c}\hfill 2xy\le \frac{x^2}{{\ell }^2}+{\ell }^2{y}^2,\ell >0.\end{array}$$

$$\begin{array}{cc}\hfill & |\Theta {\overline{\zeta }}^2-2{\eta }_0|E_{CH}{\int }_0^k|N_r^{(n+1)}-N_r^{(n)}||N_r^{(n)}-N_r^{(n-1)}|dr\hfill \\ \hfill \le & \frac{1}{6T}{E}_{CH}{\int }_0^k|N_r^{(n+1)}-N_r^{(n)}{|}^{2}dr+\frac{3}{2}{(\Theta {\overline{\zeta }}^2-2{\eta }_0)}^{2}T{\int }_0^k{E}_{CH}[\underset{0\le a\le r}{sup}|N_a^{(n)}-N_a^{(n-1)}{|}^2]dr\hfill \\ \hfill \le & \frac{1}{6}{E}_{CH}[\underset{0\le a\le k}{sup}|N_a^{(n+1)}-N_a^{(n)}{|}^2]+\frac{3}{2}{(\Theta {\overline{\zeta }}^2-2{\eta }_0)}^{2}T{\int }_0^k{E}_{CH}[\underset{0\le a\le r}{sup}|N_a^{(n)}-N_a^{(n-1)}{|}^2]dr.\hfill \end{array}$$

(16)

By (c.4), we can get

$$\begin{array}{c}\hfill E_{CH}{\int }_0^k\parallel g(N_r^{(n)})-g(N_r^{(n-1)}){\parallel }_2^{2}dr\le {\lambda }_2^2{E}_{CH}{\int }_0^k\underset{0\le s\le r}{sup}{|N_s^{(n)}-N_s^{(n-1)}|}^{2}dr.\end{array}$$

(17)

By (c.2), we can obtain

$$\begin{array}{cc}\hfill & 2E_{CH}{\int }_0^k|f(N_r^{(n)})-f(N_r^{(n-1)})||N_r^{(n+1)}-N_r^{(n)}|dr\hfill \\ \hfill & \le \frac{1}{6T}{E}_{CH}{\int }_0^k|N_r^{(n+1)}-N_r^{(n)}{|}^{2}dr+6{\lambda }_1^{2}T{E}_{CH}{\int }_0^k{\parallel N_r^{(n)}-N_r^{(n-1)}\parallel }_C^{2}dr\hfill \\ \hfill & \le \frac{1}{6}{E}_{CH}[\underset{0\le s\le k}{sup}|N_s^{(n+1)}-N_s^{(n)}{|}^2]+6{\lambda }_1^{2}T{\int }_0^k{E}_{CH}[\underset{0\le s\le r}{sup}|N_s^{(n)}-N_s^{(n-1)}{|}^2]dr.\hfill \end{array}$$

By the BDG inequality, it holds that

$$\begin{array}{cc}\hfill & 2E_{CH}[\underset{0\le s\le k}{sup}|{\int }_0^s(N_r^{(n+1)}-N_r^{(n)},(g(N_r^{(n)})-g(N_r^{(n-1)}))d{W}_r|]\hfill \\ \hfill \le & 6E_{CH}[\underset{0\le s\le k}{sup}|N_s^{(n+1)}-N_s^{(n)}|({\int }_0^k\parallel g(N_r^{(n)})-g(N_r^{(n-1)}){\parallel }_2^{2}dr{)}^{\frac{1}{2}}]\hfill \\ \hfill \le & \frac{1}{6}{E}_{CH}[\underset{0\le s\le k}{sup}|N_s^{(n+1)}-N_s^{(n)}{|}^2]+54{\lambda }_2^2{\int }_0^k{E}_{CH}[\underset{0\le s\le r}{sup}|N_r^{(n)}-N_r^{(n-1)}{|}^2]dr.\hfill \end{array}$$

(18)

By the extension of Liu’s lemma (Corollary 2), it holds that

$$\begin{array}{cc}\hfill & 2E_{CH}[\underset{0\le s\le k}{sup}|{\int }_0^s(N_r^{(n+1)}-N_r^{(n)},(h(N_r^{(n)})-h(N_r^{(n-1)}))d{C}_r|]\hfill \\ \hfill \le & 2L{E}_{CH}[\underset{0\le s\le k}{sup}|N_s^{(n+1)}-N_s^{(n)}|({\int }_0^k|h(N_r^{(n)})-h(N_r^{(n-1)}){|}^{2}dr{)}^{1/2}]\hfill \\ \hfill \le & \frac{1}{6}{E}_{CH}[\underset{0\le s\le k}{sup}|N_s^{(n+1)}-N_s^{(n)}{|}^2]+K{\lambda }_1^2{\int }_0^k{E}_{CH}[\underset{0\le s\le r}{sup}|N_r^{(n)}-N_r^{(n-1)}{|}^2]dr,\hfill \end{array}$$

(19)

where K is a constant, and we denote

$$\begin{array}{c}\hfill {\psi }^{(n)}(k)=E_{CH}[\underset{0\le a\le k}{sup}|N_a^{(n+1)}-N_a^{(n)}{|}^2],\end{array}$$

(20)

combining with (15)–(19), it holds that

$$\begin{array}{c}\hfill {\psi }^{(n)}(k)\le \frac{2}{3}{\psi }^{(n)}(k)+c{\int }_0^k{\psi }^{(n-1)}(r)dr,\end{array}$$

(21)

where $c>0$ is a positive constant, so there exists $l>0$ such that

$$\begin{array}{c}\hfill {\psi }^{(n)}(k)\le l{\int }_0^k{\psi }^{(n-1)}(r)dr.\end{array}$$

(22)

By iteration from (22), we obtain

$$\begin{array}{c}\hfill {\psi }^{(n)}(k)\le \frac{l^{(n-1)}{T}^{(n-1)}}{(n-1)!}{\psi }^{(1)}(k),\forall n>1,\forall k\in [0,T].\end{array}$$

(23)

Therefore,

$$\begin{array}{c}\hfill E_{CH}[\underset{0\le a\le T}{sup}|N_a^{(n+1)}-N_a^{(n)}{|}^2]\le \frac{l^{(n-1)}{T}^{(n-1)}}{(n-1)!}{\psi }^{(1)}(k),\forall n>1.\end{array}$$

(24)

By (24), it is clear that $\left\{N_k^{(n)}\right\}$ is a Cauchy sequence $L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))$. □

Lemma 5.

Assume the conditions $(c.1),(c.2),(c.3),(c.4),(c.5),(c.6)$ hold. Then, the sequence $\left\{N_k^{(n)}\right\}$ is bounded in $I^2(0,T;Z)$.

Proof. 

By applying Itô-Liu’s formula to $|N_k^{(n)}{|}^2$ with $n\ge 2$, we can obtain

$$\begin{array}{cc}\hfill & E_{CH}{|N_k^{(n)}|}^2\hfill \\ \hfill =& 2E_{CH}{\int }_0^k\langle -\frac{\partial N_r^{(n)}}{\partial \theta },N_r^{(n)}\rangle dr-2{\int }_0^k(\eta (r,\theta )N_r^{(n-1)},N_r^{(n)})dr\hfill \\ \hfill & -\Theta {\overline{\zeta }}^2{E}_{CH}{\int }_0^k|N_r^{(n)}{|}^{2}dr+E_{CH}{|N_0|}^2+2E_{CH}{\int }_0^k(f(N_r^{(n-1)}),N_r^{(n)})dr\hfill \\ \hfill & +\Theta {\overline{\zeta }}^2{E}_{CH}{\int }_0^k(N_r^{(n)},N_r^{(n-1)})dr-2E_{CH}{\int }_0^k(f(N_r^{(n-1)}),N_r^{(n-1)})dr\hfill \\ \hfill & +2E_{CH}{\int }_0^k(f(N_r^{(n-1)}),N_r^{(n-1)})dr+E_{CH}{\int }_0^k{\parallel g(N_r^{(n-1)})\parallel }_2^{2}dr.\hfill \end{array}$$

(25)

Thus,

$$\begin{array}{cc}\hfill & -2E_{CH}{\int }_0^k(f(N_r^{(n-1)}),N_r^{(n-1)})dr-E_{CH}{\int }_0^k{\parallel g(N_r^{(n-1)})\parallel }_2^{2}dr\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+2E_{CH}{\int }_0^k|f(N_r^{(n-1)})|(|N_r^{(n)}|+|N_r^{(n-1)}|)dr+|\Theta {\overline{\zeta }}^2-2{\eta }_0|E_{CH}{\int }_0^k|N_r^{(n)}||N_r^{(n-1)}|dr.\hfill \end{array}$$

(26)

Here, we see that

$$\begin{array}{cc}\hfill & 2E_{CH}{\int }_0^k|f(N_r^{(n-1)})|(|N_r^{(n)})|+|N_r^{(n-1)})|)dr\hfill \\ \hfill \le & 2{\lambda }_1{E}_{CH}{\int }_0^k\parallel N_r^{(n-1)}{\parallel }_{\mathcal{C}}(|N_r^{(n)}|+|N_r^{(n-1)})|)dr\hfill \\ \hfill \le & {\lambda }_1{E}_{CH}{\int }_0^k[\parallel N_r^{(n-1)}{\parallel }_{\mathcal{C}}^2+(|N_r^{(n)}|+|N_r^{(n-1)}{|)}^2]dr\hfill \\ \hfill \le & T{\lambda }_1{E}_{CH}(\underset{0\le a\le T}{sup}|N_a^{(n-1)}{|}^2)+2{\lambda }_{1}T[E_{CH}(\underset{0\le a\le T}{sup}|N_a^{(n)}{|}^2)+E_{CH}(\underset{0\le a\le T}{sup}|N_a^{(n-1)}{|}^2)]\hfill \\ \hfill =& T{\lambda }_1\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2+2{\lambda }_{1}T[\parallel N_k^{(n)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2+\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2],\hfill \end{array}$$

(27)

and

$$\begin{array}{cc}\hfill & |\Theta {\overline{\zeta }}^2-2{\eta }_0|E_{CH}{\int }_0^k|N_r^{(n)}||N_r^{(n-1)}|dr\hfill \\ \hfill \le & \frac{1}{2}|\Theta {\overline{\zeta }}^2-2{\eta }_0|T[E_{CH}(\underset{0\le a\le T}{sup}|N_a^{(n)}{|}^2)+E_{CH}(\underset{0\le a\le T}{sup}|N_a^{(n-1)}{|}^2)]\hfill \\ \hfill =& \frac{1}{2}|\Theta {\overline{\zeta }}^2-2{\eta }_0|T[\parallel N_k^{(n)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2+\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2].\hfill \end{array}$$

(28)

By (c.6), the following inequalities hold:

$$\begin{array}{cc}\hfill & \varsigma {\int }_0^k{E}_{CH}{\parallel N_r^{(n-1)}\parallel }^{2}dr\hfill \\ \hfill \le & -2E_{CH}{\int }_0^k(f(N_r^{(n-1)}),N_r^{(n-1)})dr-E_{CH}{\int }_0^k\parallel g(N_r^{(n-1)}){\parallel }_2^{2}dr+|\lambda |T\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2\hfill \\ & +{\int }_0^k\chi (r)e^{-\rho r}dr.\hfill \end{array}$$

(29)

We can see that, owing to the subexponential growth and continuity of the $\chi (k)e^{-\rho k}$, $\forall k\in {\mathbb{R}}^{+}$, there exists a positive constant $\overline{\chi }$ such that $\chi (k)e^{-\rho k}\le \overline{\chi }$. Combining with (26)–(29), we can deduce the following inequality.

$$\begin{array}{cc}\hfill & \varsigma {\int }_0^k{E}_{CH}{\parallel N_r^{(n-1)}\parallel }^{2}dr\hfill \\ \hfill \le & -2E_{CH}{\int }_0^k(f(N_r^{(n-1)}),N_r^{(n-1)})dr-E_{CH}{\int }_0^k\parallel g(N_r^{(n-1)}){\parallel }_2^{2}dr+|\lambda |T\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2\hfill \\ & +{\int }_0^k\chi (r)e^{-\rho r}dr,\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+T{\lambda }_1\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2+(\frac{1}{2}|\Theta {\overline{\zeta }}^2-2{\eta }_0|+2{\lambda }_1)T[\parallel N_k^{(n)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2\hfill \\ & +\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2]+|\lambda |T\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2+T\overline{\chi },\hfill \\ \hfill =& E_{CH}|N_0{|}^2+(3T{\lambda }_1+\frac{1}{2}|\Theta {\overline{\zeta }}^2-2{\eta }_0|+|\lambda |T)\parallel N_k^{(n-1)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2\hfill \\ & +(\frac{1}{2}|\Theta {\overline{\zeta }}^2-2{\eta }_0|+2{\lambda }_1)T\parallel N_k^{(n)}{\parallel }_{L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))}^2+T\overline{\chi }.\hfill \end{array}$$

(30)

Because $\left\{N^{(n)}\right\}$ is convergent in $L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))$, which leads to the boundedness of $\left\{N^{(n)}\right\}$. Thus, there exists a constant $l^{\prime }$ such that

$${\int }_0^k{E}_{CH}{\parallel N_r^{(n-1)}\parallel }^{2}dr\le l^{\prime },$$

and Lemma 5 is proved. □

Lemma 6.

Assume the conditions (c.1)–(c.6), and $\Theta \overline{\zeta }=0$ hold. Then, there exists a unique process $N_k\in Q$ such that

$$N_k=N_0+{\int }_0^k[\frac{\partial N_r}{\partial \theta }+f_1(r)]dr+{\int }_0^{k}h(r)d{C}_r+M(k),a.s.,\forall k\in [0,T],$$

where $f_1,h\in I^2(0,T;H)$, $N_0\in L^2(\Gamma \times \Omega ,\mathcal{L}$${}_0\otimes \mathcal{F}$${}_0,\mathcal{M}\times P;H)$ and $M(k)$ is an H-valued continuous, square integrable $\mathcal{F}$${}_k$-martingale.

Proof. 

See the Refs. [26,32] and, similar to Proposition 3.1 in [22], it is easy to get the Lemma 6. □

Next, we give the proof of the existence of a solution to the problem (5).

Theorem 2.

(Existence) Assume (c.1)–(c.4) and (c.6) hold. Then, for each $N_0\in Q$, there exists a unique solution of Equation (5) in $Q.$

Proof. 

The family ${\Theta }_1(k,·):Z\to Z^{\prime }$, defined as ${\Theta }_1(k,N_k)=-\frac{\partial N_k}{\partial \theta }-(\Theta {\overline{\zeta }}^2/2)N_k$, satisfies the assumptions in Lemma 6. Combining with (c.1), as a result, (10)–(12) have a unique solution $N_k^{(1)}\in Q$.

For $n=1$, it follows from (c.2), (c.4) and Lemma 6 that there exists a unique process $N_k^{(2)}\in Q$ of (10)–(12).

By recurrence, it holds that ${\left\{N_k^{(n)}\right\}}_{n\ge 1}\subset Q$ is a unique solution of (10)–(12).

Next, we will prove that the sequence $\left\{N_k^{(n)}\right\}$ is convergent to a process $N_k$ in Q, which will be the solution of (5).

Firstly, by Lemma 4, it holds that there exists

$$N_k\in L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))$$

such that $N_k^{(n)}\to N_k$ in $L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))$. By (c.2) and (c.4), we can obtain $f(N_k^{(n)})\to f(N_k)$, and $h(N_k^{(n)})\to h(N_k)$ in $L^2(\Gamma \times \Omega ;L^{\infty }(0,T;H)))$, $g(N_k^{(n)})\to g(N_k)$ in $L^2(\Gamma \times \Omega ;L^{\infty }(0,T;\mathfrak{L}$$(S,H))))$.

Let

$$D{N}_k=\frac{\partial N_k}{\partial k}+\frac{\partial N_k}{\partial \theta }.$$

So

$$\begin{array}{cc}\hfill D{N}^{(n)}(k,\theta )=& -\frac{\Theta {\overline{\zeta }}^2}{2}{N}_k^{(n)}dk-\eta (k,\theta )N_k^{(n-1)}dk+\frac{\Theta {\overline{\zeta }}^2}{2}{N}_k^{(n-1)}dk\hfill \\ & +f(k,N_k^{(n-1)})dk+g(k,N_k^{(n-1)})d{W}_k+h(k,N_k^{(n-1)})d{C}_k.\hfill \end{array}$$

Through the previous analysis, it is easily to get that

$$\parallel D{N}^{(n)}{\parallel }_{Z^{\prime }}\le l\le \infty .$$

In addition, according to Lemma 5, $\left\{N_k^{(n)}\right\}$ has a subsequence which is weakly convergent in $I^2(0,T;Z)$. Owing to $N_k^{(n)}\to N_k$ in $L^2(\Gamma \times \Omega ;\mathcal{C}(0,T;H))$, it holds that $N_k^{(n)}\to N_k$ weakly in $I^2(0,T;Z)$ (denoted by $N_k^{(n)}\rightharpoonup N_k$), and $D{N}^{(n)}\rightharpoonup lin{L}^2(\Gamma \times \Omega \times (0,T);Z^{\prime }).$

Owing to the continuity of the differential operator D, $DN=l$. Combining with the proof of uniqueness of Theorem 1, we complete the proof of Theorem 2. □

3.3. Stability Theorem

Theorem 3.

Assume $(c.1),(c.2),(c.3),(c.4),(c.5),(c.6)$ and $\varsigma /c^2-\lambda -\Theta {\overline{\zeta }}^2+2{\eta }_0>0$ hold. If $N_k$ is a global strong solution to Equation (5), then there exist constants $\kappa >0$, $D>0$ such that

$$\begin{array}{c}\hfill E_{CH}{|N_k|}^2\le D·e^{-\kappa k},\forall k\ge 0.\end{array}$$

(31)

Proof. 

Take small enough $ϵ>0$ such that $\rho -ϵ>0$. By Itô-Liu’s formula, it holds that

$$\begin{array}{cc}\hfill & e^{(\rho -ϵ)k}|N_k{|}^2-{|N_0|}^2\hfill \\ \hfill =& (\rho -ϵ){\int }_0^k{e}^{(\rho -ϵ)r}{|N_r|}^{2}dr+2{\int }_0^k{e}^{(\rho -ϵ)r}\langle -\frac{\partial N_k}{\partial \theta }\rangle dr-\eta (r,\theta )(N_r,N_r)dr\hfill \\ \hfill & +2{\int }_0^k{e}^{(\rho -ϵ)r}(f(r,N_r),N_r)dr+2{\int }_0^k{e}^{(\rho -ϵ)r}(g(r,N_r)d{W}_r,N_r)\hfill \\ \hfill & +2{\int }_0^k{e}^{(\rho -ϵ)r}(h(r,N_r)d{C}_r,N_r)+{\int }_0^k{e}^{(\rho -ϵ)r}{\parallel g(r,N_r)\parallel }_2^{2}dr.\hfill \end{array}$$

(32)

By Lemma 2, it follows that

$$E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(g(r,N_r)d{W}_r,N_r)=0,k\in {\mathbb{R}}^{+}.$$

and

$$E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(h(r,N_r)d{C}_r,N_r)=0,k\in {\mathbb{R}}^{+}.$$

Therefore,

$$\begin{array}{cc}\hfill & e^{(\rho -ϵ)k}{E}_{CH}{|N_k|}^2\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+(\rho -ϵ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr-2E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}\langle \frac{\partial N}{\partial \theta },N_r\rangle dr\hfill \\ \hfill & -2{\eta }_0{E}_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(N_r,N_r)dr+2E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(f(r,N_r),N_r)dr+E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}{\parallel g(r,N_r)\parallel }_2^{2}dr\hfill \\ \hfill =& E_{CH}|N_0{|}^2+(\rho -ϵ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}|N_r{|}^{2}dr+E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}{\parallel g(r,N_r)\parallel }_2^{2}dr\hfill \\ \hfill & -2{\eta }_0{E}_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr+2E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(f(r,N_r),N_r)dr+E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}{({\int }_0^{\Theta }\zeta (r,\theta )N_{r}da)}^{2}dr\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+(\rho -ϵ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}|N_r{|}^{2}dr+E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}\parallel g(r,N_r){\parallel }_2^{2}dr-2{\eta }_0{\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr\hfill \\ & +2E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(f(r,N_r),N_r)dr+{\int }_0^k{e}^{(\rho -ϵ)r}({\int }_0^{\Theta }{\zeta }^2(r,\theta )d\theta E_{CH}{\int }_0^{\Theta }{N}_r^{2}d\theta )dr.\hfill \end{array}$$

By condition (c.6) and (3) yield

$$\begin{array}{cc}\hfill & e^{(\rho -ϵ)k}{E}_{CH}{|N_k|}^2\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+(\rho -ϵ+\lambda ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}|N_r{|}^{2}dr-\varsigma {\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{\parallel N_r\parallel }^{2}dr\hfill \\ \hfill & +(\Theta {\overline{\zeta }}^2-2{\eta }_0){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr+{\int }_0^k\chi (r)e^{-ϵr}dr\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+(\rho -ϵ-\varrho ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr+{\int }_0^k\chi (r)e^{-ϵr}dr,\hfill \end{array}$$

(33)

where $\varrho =\varsigma /c^2-\lambda -\Theta {\overline{\zeta }}^2+2{\eta }_0$.

If $\rho -\varrho \le 0$, it follows immediately

$$\begin{array}{c}\hfill e^{(\rho -ϵ)k}{E}_{CH}|N_k{|}^2\le E_{CH}{|N_0|}^2+{\int }_0^k\chi (r)e^{-ϵr}dr.\end{array}$$

(34)

Then, there exists a positive constant $l(ϵ)$ such that

$$\begin{array}{c}\hfill E_{CH}|N_k{|}^2\le (E_{CH}|N_0{|}^2+l(ϵ))e^{-(\rho -ϵ)k}.\end{array}$$

(35)

If $\rho -\varrho >0$, we take small enough $ϵ>0$ such that $\rho -ϵ-\varrho >0$. Thus, by (33) and Gronwall’s lemma (Lemma 3), we can obtain

$$e^{(\rho -ϵ)k}{E}_{CH}|N_k{|}^2\le (E_{CH}|N_0{|}^2+{\int }_0^k\chi (r)e^{-ϵr}dr)e^{k(\rho -ϵ-\varrho )}.$$

So, there exists a positive constant $l(ϵ)>0$ such that

$$\begin{array}{c}\hfill E_{CH}|N_k{|}^2\le (E_{CH}|N_0{|}^2+l(ϵ))e^{-\varrho k}.\end{array}$$

(36)

Next, we will give a generalization of Theorem 3.

(c.7) There exist constants $\varsigma >0$, $\rho >0$, $\lambda \in R$, $0\le \delta <1$ and a nonnegative continuous function $\chi (k)$, $\kappa (k)$$k\in R^{+}$, such that

$$2\langle f(k,z),z\rangle +{\parallel g(k,z)\parallel }_2^2\le -{\varsigma \parallel z\parallel }^2+{\lambda |z|}^2+\chi (k)e^{-\rho k}+\kappa (k)e^{-\rho k}{|z|}^{2\delta },z\in Z.$$

Here, $\forall ϵ>0$, $\chi (k)$ satisfy $\underset{k\to \infty }{lim}\chi (k)/e^{ϵk}=0$, $\underset{k\to \infty }{lim}\kappa (k)/e^{ϵk}=0$. □

Theorem 4.

Assume $(c.1),(c.2),(c.3),(c.4),(c.5),(c.7)$ and $\varsigma /c^2-\lambda -\Theta {\overline{\zeta }}^2+2{\eta }_0>0$ hold. If $N_k$ is a global strong solution to Equation (5), there exist constants $\kappa >0$, $D>0$ such that

$$\begin{array}{c}\hfill E_{CH}{|N_k|}^2\le D·e^{-\kappa k},\forall k\ge 0.\end{array}$$

(37)

Proof. 

Similar to the proof of Theorem 3, take small enough $ϵ>0$ such that $\rho -ϵ>0$. By Itô-Liu’s formula, it holds that

$$\begin{array}{cc}\hfill & e^{(\rho -ϵ)k}|N_k{|}^2-{|N_0|}^2\hfill \\ \hfill =& (\rho -ϵ){\int }_0^k{e}^{(\rho -ϵ)r}{|N_r|}^{2}dr+2{\int }_0^k{e}^{(\rho -ϵ)r}\langle -\frac{\partial N_k}{\partial \theta }-\eta (r,\theta )N_r,N_r\rangle dr\hfill \\ \hfill & +2{\int }_0^k{e}^{(\rho -ϵ)r}(f(r,N_r),N_r)dr+2{\int }_0^k{e}^{(\rho -ϵ)r}(g(r,N_r)d{W}_r,N_r)\hfill \\ \hfill & +2{\int }_0^k{e}^{(\rho -ϵ)r}(h(r,N_r)d{C}_r,N_r)+{\int }_0^k{e}^{(\rho -ϵ)r}{\parallel g(r,N_r)\parallel }_2^{2}dr.\hfill \end{array}$$

(38)

Thus,

$$\begin{array}{cc}\hfill & e^{(\rho -ϵ)k}{E}_{CH}{|N_k|}^2\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+(\rho -ϵ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr-2E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}\langle \frac{\partial N}{\partial \theta },N_r\rangle dr\hfill \\ \hfill & -2{\eta }_0{E}_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(N_r,N_r)dr+2E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(f(r,N_r),N_r)dr+E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}{\parallel g(r,N_r)\parallel }_2^{2}dr\hfill \\ \hfill =& E_{CH}|N_0{|}^2+(\rho -ϵ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}|N_r{|}^{2}dr+E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}{\parallel g(r,N_r)\parallel }_2^{2}dr\hfill \\ \hfill & -2{\eta }_0{E}_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr+2E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(f(r,N_r),N_r)dr+E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}{({\int }_0^{\Theta }\zeta (r,\theta )N_{r}da)}^{2}dr\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+(\rho -ϵ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}|N_r{|}^{2}dr+E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}\parallel g(r,N_r){\parallel }_2^{2}dr-2{\eta }_0{\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr\hfill \\ & +2E_{CH}{\int }_0^k{e}^{(\rho -ϵ)r}(f(r,N_r),N_r)dr+{\int }_0^k{e}^{(\rho -ϵ)r}({\int }_0^{\Theta }{\zeta }^2(r,\theta )d\theta E_{CH}{\int }_0^{\Theta }{N}_r^{2}d\theta )dr.\hfill \end{array}$$

By $(c.7)$, (3), it holds that

$$\begin{array}{cc}\hfill & e^{(\rho -ϵ)k}{E}_{CH}{|N_k|}^2\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+(\rho -ϵ+\lambda ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}|N_r{|}^{2}dr-\varsigma {\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{\parallel N_r\parallel }^{2}dr\hfill \\ \hfill & +(\Theta {\overline{\zeta }}^2-2{\eta }_0){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}|N_r{|}^{2}dr+{\int }_0^k\chi (r)e^{-ϵr}dr+{\int }_0^k\kappa (r)e^{-ϵr}{E}_{CH}{|N_r|}^{2\delta }dr\hfill \\ \hfill \le & E_{CH}|N_0{|}^2+(\rho -ϵ-\varrho ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}|N_r{|}^{2}dr+{\int }_0^k\chi (r)e^{-ϵr}dr+{\int }_0^k\kappa (r)e^{-(ϵ+\delta (\rho -ϵ))r}(e^{(\rho -ϵ)r}{E}_{CH}|N_r{{|}^2)}^{\delta }dr,\hfill \end{array}$$

(39)

where $\varrho =\varsigma /c^2-\lambda -\Theta {\overline{\zeta }}^2+2{\eta }_0$.

If $\rho -\varrho \le 0$, it follows immediately

$$\begin{array}{cc}\hfill e^{(\rho -ϵ)k}{E}_{CH}{|N_k|}^2& \le E_{CH}|N_0{|}^2+{\int }_0^k\chi (r)e^{-ϵr}dr+{\int }_0^k\kappa (r)e^{-(ϵ+\delta (\rho -ϵ))r}(e^{(\rho -ϵ)r}{E}_{CH}|N_r{{|}^2)}^{\delta }dr.\hfill \end{array}$$

(40)

By the extended Gronwall-type Lemma [27], it holds that

$$\begin{array}{cc}\hfill e^{(\rho -ϵ)k}{E}_{CH}{|N_k|}^2\le & [(E_{CH}|N_0{|}^2+{\int }_0^k\chi (r)e^{-ϵr}dr{{)}^{1-\delta }+(1-\delta ){\int }_0^k\kappa (r)e^{-(ϵ+\delta (\rho -ϵ))r}dr]}^{\frac{1}{1-\delta }}.\hfill \end{array}$$

(41)

Then, there exists a positive constant $l(ϵ)$ such that

$$\begin{array}{c}\hfill E_{CH}|N_k{|}^2\le (E_{CH}|N_0{|}^2+l(ϵ))e^{-(\rho -ϵ)k}.\end{array}$$

(42)

If $\rho -\varrho >0$, we take small enough $ϵ>0$ such that $\rho -ϵ-\varrho >0$. So, by (39), we can obtain

$$\begin{array}{cc}\hfill e^{(\rho -ϵ)k}{E}_{CH}{|N_k|}^2\le & \left(E_{CH}|N_0{|}^2+{\int }_0^k\chi (r)e^{-ϵr}dr+{\int }_0^k\kappa (r)e^{-ϵr}{E}_{CH}{|N_r|}^{2\delta }dr\right)\hfill \\ & +(\rho -ϵ-\varrho ){\int }_0^k{e}^{(\rho -ϵ)r}{E}_{CH}{|N_r|}^{2}dr.\hfill \end{array}$$

By the Gronwall’s lemma (Lemma 3), we can obtain

$$e^{(\rho -ϵ)k}{E}_{CH}{|N_k|}^2\le \left(E_{CH}|N_0{|}^2+{\int }_0^k\chi (r)e^{-ϵr}dr+{\int }_0^k\kappa (r)e^{-ϵr}{E}_{CH}{|N_r|}^{2\delta }dr\right)e^{k(\rho -ϵ-\varrho )}.$$

Thus, it follows that

$$E_{CH}|N_k{|}^2\le [(E_{CH}|N_0{|}^2+{\int }_0^k\chi (r)e^{-ϵr}dr+{\int }_0^k\kappa (r)e^{-ϵr}(E_{CH}|N_r{|}^2{)}^{\delta }dr]e^{-\varrho k}.$$

By the extended Gronwall-type Lemma again, it holds that

$$\begin{array}{cc}\hfill E_{CH}{|N_k|}^2\le & e^{-\varrho k}[(E_{CH}|N_0{|}^2+{\int }_0^k\chi (r)e^{-ϵr}dr{{)}^{1-\delta }+(1-\delta ){\int }_0^k\kappa (r)e^{-ϵr}dr]}^{\frac{1}{1-\delta }}\hfill \\ \hfill \le & D(ϵ)e^{-\varrho k}.\hfill \end{array}$$

(43)

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Remark 5.

For the finite dimensional case, such as one dimensional case, we take $Z=H=\mathbb{R}$, $S=\mathbb{R}$. As a result, $W_k$ is a one-dimensional Wiener process, $G=1$ and $C_k$ is a one-dimensional Liu process.

Remark 6.

Different from existence, uniqueness and exponential stability of stochastic age-dependent population equation based on probability theory of additive measure [3,4], uncertain stochastic hybrid age-dependent population equations are more complex in terms of dealing with conditions and calculation of existence, uniqueness and exponential stability, such as the conditions of Lemmas 4–6, Theorems 1–4. In addition, we use the Itô-Liu formula, the Liu lemma (Lemma 1), the extended Liu’s lemma (Corollary 2), etc. And these conclusions are all obtained under subadditive measures. Specifically, Refs. [3,4] can be seen as the special case of the model proposed in this paper.

4. Conclusions

The main objective of this study is to propose a new and innovative type of hybrid APDS. In doing so, several significant contributions have been made. Firstly, we redefine the Liu integral in a more comprehensive manner, specifically in terms of its mean square sense. This refined definition enhances our understanding of the integral and its application in the context of the hybrid APDS. Moreover, this study extends the Liu’s lemma and the Itô-Liu formula, expanding their scope and applicability to incorporate the hybrid APDS. By extending these fundamental mathematical tools, we are able to provide a more comprehensive framework for analyzing and understanding the dynamics of the hybrid APDS. Furthermore, an essential aspect of this study is confirming the existence, uniqueness, and stability of the hybrid APDS based on subadditive measures. Through rigorous mathematical analysis and proof, we establish the fundamental properties and characteristics of the hybrid APDS based on subadditive measure, providing a solid theoretical foundation for future research. Overall, this study significantly advances our understanding of the hybrid APDS by redefining key mathematical concepts, extending established formulas, and establishing its essential properties. These contributions lay the groundwork for future investigations and enable us to delve deeper into the dynamics and behavior of the hybrid APDS based on subadditive measure in various models.