Modified square root process

Question

I am dealing with the following stochastic differential equation (SDE)

$ \begin{cases} dS_t &= \mu S_t dt + \sigma_1 S_tdW^1_t\\dG_t &= kS_t(\alpha - G_t)dt + \sigma_2\sqrt{G_tS_t}dW^2_t \end{cases} $

with $W^1$ and $W^2$ two independent Brownian motions and all the other parameters positive constants. The first component is a GBM and thus remains positive at every time. I'm interested in whether or not the second one may hit zero or not and if it does, what happen. It remains there or get reflected?

I know the classic square root process $ dr_t = k(\alpha - r_t)dt + \sigma\sqrt{r_t}dW_t $ remains strictly positive if $ 2k\alpha > \sigma^2$ (Feller condition) and nonnegative if $ 0< 2k\alpha \leq \sigma^2$, zero being a reflecting state in the latter case.

The heuristic reasoning that, at $G=0$, the equation reduces to

$ dG_t= k\alpha S_t dt$

and it is thus pushed upward makes me believe that zero is a reflecting barrier also in this case but I don't know how to formalize it.

Answer 1

A slightly less heuristic argument that agrees with your conclusion:

Let $k_t=kS_t$ and $\sigma_t=\sigma_2\sqrt{S_t}$. Then your equation is $$dG_t = k_t(\alpha-G_t)dt + \sigma_t\sqrt{G_t}dW_t$$ which looks like the classic square root process except that now $k$ and $\sigma$ are stochastic. The Feller condition $2k\alpha>\sigma^2_2$ becomes $2kS_t\alpha>\sigma_2^2S_t$, and $S_t$ cancels out, so it's the same condition as before.