Let $B(t)$ be a 1-dimensional Brownian motion. Consider the Ito integral form $$dx=v\,dt+dB\\dv = -κx\,dt-\alpha\,v\,dt+h\,dB$$ where $\kappa$, $\alpha$ and $h$ are positive constants. I would like to prove the existence and uniqueness of this 2D SDE. I think that the reference could be the following theorem.
Theorem. For the SDE $$ \mathrm{d} X=G(t, X(t)) \mathrm{d} t+H(t, X(t)) \mathrm{d} W(t), \quad X\left(t_{0}\right)=X_{0}, $$ assume the following hold.
(1) Both $G(t, x)$ and $H(t, x)$ are continuous on $(t, x) \in\left[t_{0}, T\right] \times \mathbb{R}$.
(2) The coefficient functions $G$ and $H$ satisfy the Lipschitz condition $$ |G(t, x)-G(t, y)|+|H(t, x)-H(t, y)| \leq K|x-y| . $$ (3) The coefficient functions $G$ and H satisfy a growth condition in the second variable, $$ |G(t, x)|^{2}+|H(t, x)|^{2} \leq K\left(1+|x|^{2}\right), $$ for all $t \in\left[t_{0}, T\right]$ and $x \in \mathbb{R}$.
Then the SDE has a strong solution on $\left[t_{0}, T\right]$ that is continuous with probability 1 and $$ \sup _{t \in\left[t_{0}, T\right]} \mathbb{E}\left[X^{2}(t)\right]<\infty, $$ and for each given Wiener process $W(t)$, the corresponding strong solutions are pathwise unique, which means that if $X$ and $Y$ are two strong solutions, then $$ \mathbb{P}\left[\sup _{t \in\left[t_{0}, T\right]}|X(t)-Y(t)|=0\right]=1 . $$
(I took it from Dunbar, S.R. Mathematical Modeling in Economics and Finance: Probability, Stochastic Processes, and Differential Equations; Vol. 49, American Mathematical Soc., 2019.)
In my case I have that $X=(x,v)$ and $G$, for example, is the matrix: $$G(t,X) = \begin{bmatrix} 0 & 1 \\-\kappa & -\alpha\end{bmatrix}$$ But how can I prove the conditions (1)-(3) of the theorem above?