So I have two 1D diffusion processes, $$dX_t = \mu_X dt + b_X dB_t$$ $$dY_t = \mu_Ydt + b_Y dB_t.$$
I want to write these as a 2D process $Z_t = (X_t, Y_t)$.
Now, the dynamics are piecewise in the sense that only one of $X_t$ and $Y_t$ occurs at one time, and the choice depends on a function $f(X_t,Y_t)$.
Specifically if $f>0$, then $X_t$ is occurring with probability $1-\epsilon$, and $Y_t$ is occurring with probability $\epsilon$. If $f<0$ then $Y_t$ occurs with probability $1-\epsilon$ and $X_t$ occurs with probability $\epsilon$.
When each is occurring, it is very simple: $X_t$ ($Y_t$) has constant drift $p_x$ ($p_y$) and both have variance $1$.
So I defined
\begin{equation} \mu_X = \begin{cases} (1-\epsilon)p_x &\text{ if } f<0\\ \epsilon p_x &\text{ if }f>0 \end{cases} \end{equation}
\begin{equation} b_X = \begin{cases} (1 -\epsilon) &\text{ if } f<0\\ \epsilon &\text{ if }f>0 \end{cases} \end{equation}
The process $Y$ is defined similarly: \begin{equation} \mu_Y = \begin{cases} \epsilon p_y &\text{ if } f<0\\ (1-\epsilon)p_y &\text{ if }f>0 \end{cases} \end{equation}
\begin{equation} b_Y= \begin{cases} \epsilon &\text{ if } f<0\\ (1-\epsilon) &\text{ if }f>0 \end{cases} \end{equation}
To get these, I assumed that when a process is occurring with probability $1-\epsilon$, then you can simply scale the full (i.e. occur with probability 1) transition density $\mathcal{N}(\mu t, \sigma^2 t)$ to $\mathcal{N}(\mu t (1-\epsilon), \sigma^2 t(1-\epsilon))$. If this is wrong, please correct me.
Now, I'm not sure what the covariance structure is other than the fact that only one of the two processes occurs at a time.
My guess was to simply combine these into the 2D case with no correlation, but it doesn't seem right.
\begin{equation}
b_Z =
\begin{cases}
\begin{pmatrix}
(1-\epsilon)&0\\
0&\epsilon
\end{pmatrix} &\text{ if } f<0\\
\begin{pmatrix}
\epsilon&0\\
0&(1-\epsilon)
\end{pmatrix} &\text{ if } f>0
\end{cases}
\end{equation}
Should the off-diagonals be negative? Since the two processes are mutually exclusive, my (not very strong) intuition is that it should either be 0 or negative.