stochastic differential equations in matrix notation

Question

How is Matrix K determined?

$ dY_1 = \frac{-1}{2} Y_1dt - Y_2dB_t $

$ dY_2 = \frac{-1}{2} Y_2dt - Y_1dB_t $

In matrix notation, the above equations can be written as:

$ dY(t) = \frac{-1}{2}Y(t)dt + KY(t)dB_t $

where K is equal to

$\begin{pmatrix}0&&-1\\1&&0\end{pmatrix}$

Answer 1

$$\begin{pmatrix}dY_1\\dY_2\end{pmatrix}=\begin{pmatrix}-\frac{Y_1}{2}&&-Y_2\\ \frac{-1}{2} Y_2&&-Y_1\end{pmatrix}\begin{pmatrix}dt\\dB_t\end{pmatrix}$$

Answer 2

We have $$\begin{pmatrix}dY_1\\dY_2\end{pmatrix} = \begin{pmatrix}-\frac{1}{2}Y_1 + 0Y_2\\0Y_1 -\frac{1}{2}Y_2\end{pmatrix}dt + \begin{pmatrix}0Y_1 - 1Y_2\\ -1Y_1 + 0Y_2\end{pmatrix}dB_t $$ $$= \begin{pmatrix}-\frac{1}{2} & 0\\ 0 & -\frac{1}{2}\end{pmatrix}Ydt + \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}YdB_t,$$

so $K = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$. I assume you switched a sign somewhere.