Distribution of $ad-bc$ I'm interested in the stochastic process
$$f_t=(ad-bc)_t$$
where $(a,b,c,d)_t$ is governed by the following transition rules:
$$\begin{align}(a,b,c,d) \rightarrow \begin{cases}
(a+1,b,c,d) \;\;\;\; \text{ with probability }p_x &\text{ if }ad>bc\\
(a,b,c+1,d) \;\;\;\; \text{ with probability }1-p_x &\text{ if }ad>bc\\
(a,b+1,c,d) \;\;\;\; \text{ with probability }p_y &\text{ if }ad\leq bc\\
(a,b,c,d+1) \;\;\;\; \text{ with probability }1-p_y &\text{ if }ad\leq bc\\
\end{cases}
\end{align}$$
with initial state $(1,1,1,1)$.
I want to compute some quantities relating to $f_t$. In particular its distribution at  time $t$, $P_t$, and (I think equivalently), the probability at a given time that $f_t$ changes sign. Maybe also $n$-step transition probability densities would be useful for that goal.
Here are some thoughts:  The transitions of $f_t$ derived from the above would be
$$ f_{t+1} = \begin{cases}
f_t + d_t \;\;\;\; \text{ with probability } p_x &\text{ if } f_t > 0\\
f_t - b_t \;\;\;\; \text{ with probability } 1- p_x &\text{ if } f_t>0\\
f_t - c_t \;\;\;\; \text{ with probability } p_y &\text{ if } f_t \leq 0\\
f_t + a_t \;\;\;\; \text{ with probability } 1-p_y &\text{ if } f_t \leq 0\\
\end{cases}
$$
An attempt at a recursive relation for its distribution $P_t$ at time $t$:
\begin{align}
P_t(f_t) &= P(f_{t-1} > 0) \big[ P_{t-1}(f_t-d_{t-1})p_x + P_{t-1}(f_t+b_{t-1})(1-p_x)\big] \\
&+ P(f_{t-1} \leq 0) \big[ P_{t-1}(f_t+c_{t-1})p_y + P_{t-1}(f_t-a_{t-1})(1-p_y)\big]
\end{align}
For starters, $P_0(f) = \delta(0)$ since we assume $(a,b,c,d)$ starts at $(1,1,1,1)$. Then, $P_1(1) = (1-p_y)$ and $P_1(-1) = p_y$. Worst case, it's possible to start from $(1,1,1,1)$ and compute all the possible states in $t$ steps, so I know this must be a tractable problem.
Any input would be valued, thanks very much.
 A: Although not a full solution, here are some thoughts that may help along the way:
An elementary approach would be to formalize the problem and introduce the probability space
$$ \Omega = (\mathbb{N}^4)^\mathbb{N}, $$
i.e. the space of sequences $(a_t, b_t, c_t, d_t)_{t\in\mathbb{N}}$, so that the transition rules define a probability measure $\mathbb{P}$ on the power set $\mathcal{P}(\Omega)$.
Then each $f_t$, $t\in\mathbb{N}$, is a random variable from $\Omega \to \mathbb{Z}$ (equivalently, $f$ is a stochastic process from $\Omega \times \mathbb{N} \to \mathbb{Z}$).
Regarding the distribution of $f_t$ we may then write, for every $k\in\mathbb{Z}$,
\begin{align}
\mathbb{P}(f_t = k) = \mathbb{P}(&\{f_{t-1} = k - d_{t-1} \text{ and } a_t = a_{t-1}+1 \text{ and } f_{t-1} > 0\}\\ 
&\cup \{f_{t-1} = k + b_{t-1} \text{ and } c_t = c_{t-1}+1 \text{ and } f_{t-1} > 0\}\\
&\cup \{f_{t-1} = k + c_{t-1} \text{ and } b_t = b_{t-1}+1 \text{ and } f_{t-1} \leq 0\}\\
&\cup \{f_{t-1} = k - a_{t-1} \text{ and } d_t = d_{t-1}+1 \text{ and } f_{t-1} \leq 0\}).
\end{align}
The unions above are clearly disjoint.
