I'm just curious about the following dynamic process:
$$ x_{t+1} = x_t\frac{y_t + \varepsilon_t}{y_t + x_t} $$ $$ y_{t+1} = y_t + \varepsilon_t $$
Is there a name for this? I'm trying to find references but I'm not sure what to search for.
Edit: for some additional context, this process arises from Bayesian updating of an unobserved parameter $b \sim \Gamma(\alpha_t,y_t)$, when $\varepsilon_t \sim \Gamma(a, b)$ is observed, with known $a$ and mean $x$. So $y_t$ is the scale hyperparameter of the prior for $b$ at time $t$ and $x_t$ is the posterior predictive mean of $\varepsilon_t$ given that prior: $\mathbb{E}[\varepsilon_t\mid \alpha_t, y_t]$.
Ultimately, I'm interested in trying to estimate the $\{y_\tau\}$ when $\{\{x_\tau\}, a\}$ is observed and the $\varepsilon_t$ are observed with error, that is $\{\tilde\varepsilon_\tau\}$ are observed, where: $$ \begin{aligned} \tilde\varepsilon_t &= \varepsilon_t + \eta_t\\ \varepsilon_t &\sim \Gamma(a,b)\\ \eta_t &\sim \text{LogNormal}(0,\sigma) \text{ or } \Gamma(a_\eta,b_\eta) \text{ or something}. \end{aligned} $$
So as part of this, I want to understand the properties of the dynamic process above.