Existence and uniqueness for SDE 2D linear system

Question

I have the following SDE $$\ddot{x} +x = \dot{B(t)}$$

with some given initial condition $(x_0,\dot{x_0})$ and where $B(t)$ is a standard Brownian motion. I can reduce it to first order by introducing the variable $y(t)= \dot{x}(t)$.

Hence the system writes

$$ d \vec{x} = A \vec{x}dt + \begin{bmatrix} 0 \\ 1\end{bmatrix} dB(t) $$

and $$A = \begin{bmatrix} 0 & 1 \\-1 & 0\end{bmatrix}$$ I want to show existence and uniqueness for such a SDE, (i.e. I want to show Lipschitz condition and linear growth), but I don't know how to do because usually I have a scalar SDE, while now I have to write it as a system of first order SDEs.

How can I move? I have never faced such a 2dimensional case.

Answer 1

In this specific example you do not need SDE theory for the stated purposes. To that end, use $y=\dot x-B$ as second state variable, then \begin{align} \dot x(t) &= y(t)+B(t)\\ \dot y(t) &= -x(t) \end{align} is an linear ODE system with continuous inhomogeneity (assuming you take a pathwise continuous realization of the Brownian motion, as is usually done).