*2.1. Existence Theorem*

Now, we have the following existence theorem

**Theorem 1.** *Let the Assumptions 1–5 be satisfied, then there exists at least one solution* (*x*, *y*) ∈ *X of the problem (1)–(3).*

**Proof.** Firstly, from the results of Lemmas 2 and 3 and Arzela–Ascoli Theorem [9] we deduce that the closure of *FQ* is a compact subset.

Let (*xn*, *yn*) ∈ *Q* be such that

$$L.i.m\_{n \to \infty}(\mathfrak{x}\_{n\prime}y\_n) = (\mathfrak{x}\_{\prime}y) \quad w.p.1.$$

where *L*.*i*.*m* denotes the limit in the mean square sense of the continuous second order process ([1,2,9]). Now,

$$\begin{split} \text{L.i.} & \mathbf{u}\_{n \to \infty} \mathbf{f}(\mathbf{x}\_{n}, y\_{n}) &= \, \left( \text{L.i.} \boldsymbol{m}\_{n \to \infty} \mathbf{F}\_{1} y\_{n}, \text{L.i.} \boldsymbol{m}\_{n \to \infty} \mathbf{F}\_{2} \mathbf{x}\_{n} \right) \\ &= \, \left( \text{L.i.} \boldsymbol{m}\_{n \to \infty} \{ \mathbf{x}\_{0} - \int\_{0}^{\tau} h\_{1}(s, y\_{n}(s)) d\mathcal{W}(s) + \int\_{0}^{t} f\_{1}(s, y\_{n}(\boldsymbol{\phi}\_{1}(s))) ds \right), \\ & \qquad \text{L.i.} \boldsymbol{m}\_{n \to \infty} \{ y\_{0} - \int\_{0}^{\eta} h\_{2}(s, x\_{n}(s)) ds + \int\_{0}^{t} f\_{2}(s, x\_{n}(\boldsymbol{\phi}\_{2}(s))) d\mathcal{W}(s) \}) \\ &= \, \left( \text{x}\_{0} - \int\_{0}^{\tau} h\_{1}(s, \text{L.i.} \boldsymbol{m}\_{n \to \infty} y\_{n}(s)) d\mathcal{W}(s) + \int\_{0}^{t} f\_{1}(s, \text{L.i.} \boldsymbol{m}\_{n \to \infty} y\_{n}(\boldsymbol{\phi}\_{1}(s))) ds, \\ & \qquad y\_{0} - \int\_{0}^{\eta} h\_{2}(s, \text{L.i.} \boldsymbol{m}\_{n \to \infty} \mathbf{x}\_{n}(s)) ds + \int\_{0}^{t} f\_{2}(s, \text{L.i.} \boldsymbol{m}\_{n \to \infty} \mathbf{x}\_{n}(\boldsymbol{\phi}\_{2}(s))) d\mathcal{W}(s) \} \end{split}$$

$$\begin{aligned} &= \quad (\mathfrak{x}\_0 - \int\_0^\tau h\_1(\mathfrak{s}, \mathfrak{y}(\mathfrak{s}))dW(\mathfrak{s}) + \int\_0^t f\_1(\mathfrak{s}, \mathfrak{y}(\mathfrak{\phi}\_1(\mathfrak{s})))ds, \\ &\quad \mathfrak{y}\_0 - \int\_0^\eta h\_2(\mathfrak{s}, \mathfrak{x}(\mathfrak{s}))ds + \int\_0^t f\_2(\mathfrak{s}, \mathfrak{x}(\mathfrak{\phi}\_2(\mathfrak{s})))dW(\mathfrak{s})) \\ &= \quad (F\_1 \mathfrak{y}, F\_2 \mathfrak{x}) = F(\mathfrak{x}, \mathfrak{y}). \end{aligned}$$

Applying stochastic Lebesgue dominated convergence Theorem the operator *F* : *Q* → *Q* is continuous.

Finally, applying Schauder Fixed Point Theorem [9], we can deduce that there exists at least one solution (*x*, *y*) ∈ *Q* of the problem (1)–(3) such that *x*, *y* ∈ *C*([0, *T*], *L*2(Ω)).
