I am stuck on a problem involving two quantities, $R_\alpha(t)$ and $R_\beta(t)$, whose evolution over time depends on the price of an asset $S(t)$. Suppose that $S(t)$ follows a geometric brownian motion:
$$dS(t)=\mu S(t)dt+\sigma S(t)dW_t$$
We have that: $$\begin{align} d\log(R_\alpha)&= \begin{cases} -\frac{\gamma}{\gamma{}S(t) + S(t)} & S(t) > \gamma^{-1}\frac{R_\beta(t)}{R_\alpha(t)} \\ 0 & \gamma\frac{R_\beta(t)}{R_\alpha(t)} \leq S(t) \leq \gamma^{-1}\frac{R_\beta(t)}{R_\alpha(t)} \\ -\frac{1}{\gamma{}S(t) + S(t)} & S(t) < \gamma\frac{R_\beta(t)}{R_\alpha(t)} \end{cases},\\\\ d\log(R_\beta)&= \begin{cases} \frac{1}{\gamma{}S(t) + S(t)} & S(t) > \gamma^{-1}\frac{R_\beta(t)}{R_\alpha(t)} \\ 0 & \gamma\frac{R_\beta(t)}{R_\alpha(t)} \leq S(t) \leq \gamma^{-1}\frac{R_\beta(t)}{R_\alpha(t)} \\ \frac{\gamma}{\gamma{}S(t) + S(t)} & S(t) < \gamma\frac{R_\beta(t)}{R_\alpha(t)} \end{cases},\\\\ g(t) = \frac{R_\beta(t)}{R_\alpha(t)} \implies d\log(g(t))&= \begin{cases} \frac{1}{S(t)} & S(t) > \gamma^{-1}g(t) \\ 0 & \gamma{}g(t) \leq S(t) \leq \gamma^{-1}g(t) \\ \frac{1}{S(t)} & S(t) < \gamma{}g(t) \end{cases}, \end{align}$$ with $0<\gamma\leq1$, and $\frac{R_\beta(0)}{R_\alpha(0)}=S(0)=1$. To be clear, $R_\alpha(t)$ and $R_\beta(t)$ are dependent on time only through the price $S(t)$.
For example, consider $\gamma = 0.9$ and simple initial values $R_\alpha(0), R_\beta(0), S(0)=1$. If the price then moved up to $S(\Delta{}t)=1.5$, $R_\alpha$ would decrease (and $R_\beta$ would increase):
$$\begin{align} \log(\frac{R_\alpha(\Delta{}t)}{R_\alpha(0)})&= \int_1^{1.5}{ \begin{cases} -\frac{\gamma}{\gamma{}S(t) + S(t)} & S(t) > 0.9^{-1} \\ 0 & 0.9 \leq S(t) \leq 0.9^{-1} \\ -\frac{1}{\gamma{}S(t) + S(t)} & S(t) < 0.9 \end{cases}}dS(t) \\\\ &\implies \log(\frac{R_\alpha(\Delta{}t)}{R_\alpha(0)}) \approx -0.1422 \end{align}$$
Is it possible to get an expression for $R_\alpha(t)$ and $R_\beta(t)$ in terms of only $S(t)$'s parameters? It seems like there should be an exploitable symmetry but after a few weeks of head-scratching, I am not sure how to take it any further. My end goal is to get their expectations at some terminal time $T$.