Solve SDE with Dirac delta

Question

Let $B(t)$ be a 1-dimensional Brownian motion. Consider the scalar linear stochastic equation
$$
\ddot{x}(t)+\kappa x(t)=h \dot{B}(t) \quad \text{ on } t \geq 0,
$$
where $h$ is a positive constant and $\kappa$ is a real function.

By introducing the new variable $y(t)=\dot{x}(t)$, the corresponding Itô equation is
$$
d\left[\begin{array}{l}
x(t) \\
y(t)
\end{array}\right]=A\left[\begin{array}{l}
x(t) \\
y(t)
\end{array}\right] d t+\left[\begin{array}{l}
0 \\
h
\end{array}\right] d B(t)\quad \quad (*)
$$
where
$$
A=\left[\begin{array}{cc}
0 & 1 \\
-\kappa & 0
\end{array}\right]
$$
Given any initial value $(x(0), y(0))=\left(x_{0}, y_{0}\right) \in R^{2}$, we know that equation (*) has the unique solution
$$
\left[\begin{array}{l}
x(t) \\
y(t)
\end{array}\right]=e^{A t}\left[\begin{array}{l}
x_{0} \\
y_{0}
\end{array}\right]+\int_{0}^{t} e^{A(t-s)}\left[\begin{array}{l}
0 \\
h
\end{array}\right] d B(s)
$$
Is it possible to still calculate the solution of (*) if we add to it a Dirac delta? That is:
$$
d\left[\begin{array}{l}
x(t) \\
y(t)
\end{array}\right]=A\left[\begin{array}{l}
x(t) \\
y(t)
\end{array}\right] d t+\left[\begin{array}{l}
0 \\
h
\end{array}\right] d B(t)+\left[\begin{array}{l}
\kappa_2 \\
\kappa_1
\end{array}\right]d\Theta(t)\quad \quad (*)'
$$
where $\Theta$ is the Heaviside step function with the distributional derivative $\dot \Theta=\delta$ and $\kappa_1$, $\kappa_2$ are two real numbers.

Answer 1

Sketch:

1. replace the Brownian motion $B$ by the semi martingale $$ X_t:=\int_0^t\begin{bmatrix}0\\h\end{bmatrix}dB_s+\int_0^t\begin{bmatrix}\kappa_2\\\kappa_1\end{bmatrix}d\Theta(s)=\begin{bmatrix}0\\h\end{bmatrix}B_t+\begin{bmatrix}\kappa_2\\\kappa_1\end{bmatrix}\Theta(t) $$ whose quadratic variation should be $$ [X^1,X^1]_t=\kappa_2^2\,\Theta(t)\,,\quad[X^1,X^2]_t=\kappa_1\kappa_2\,\Theta(t)\,,\quad[X^2,X^2]_t=h^2\,t+\kappa_1^2\,\Theta(t)\,. $$ 2. use Ito's formula to check that $$ \begin{bmatrix}x(t)\\y(t)\end{bmatrix}=e^{At}\begin{bmatrix}x_0\\y_0\end{bmatrix}+\int_0^te^{A(t-s)}\,dX_s $$ solves the SDE $(*)'\,,$ that is: $$ d\begin{bmatrix}x(t)\\y(t)\end{bmatrix}=A\,\begin{bmatrix}x(t)\\y(t)\end{bmatrix}\,dt+dX_t\,. $$ I think in step 2 all we need to know is that the quadratic covariation of the continuous process $e^{At}$ with the stochastic $dX$-integral is zero. This makes the proof identical to the Brownian motion case.