Consider we have a vector of observed data points $\mathbf{y}$ with $n$ elements. Each consecutive data point in $\mathbf{y}$ represents an observation made at a specific time $t$, let's say that at $t=1$ we have the first element $y(1)$, at $t=2$ we have the second element $y(2)$ and so on until $y(n)$ at $t=n$.
Let's assume we can describe the evolution of the data in the following way:
$y_{t+1} = y_t + c\cdot (y_t)^a + x_t$
where $c, a\in \mathbb{R}_{\ge 0}$ and $x_t$ is i.i.d. $\mathcal{N}(\mu,\sigma^2)$.
First, suppose $\mu$ and $\sigma$ are known, what kind of approach should I use to compute or at least estimate the values for $c$ and $a$ ?
Second, now imagine we don't know $\mu$ and $\sigma$, how could i estimate them having $c$ and $a$ as unknowns still ? Really the major question for me here is how to deal with or solve functions that have random variables in them, also can this be said to be a differential equation with a random parameter.
I would appreciate any help on this, also if someone can point me in the right direction