Please note edits after original post changing the specific form of the setup
Let's say we have a stochastic differential equation:
$$
\mathrm{d}S_t = |S^\beta| {(\mu \mathrm{d}t + \sigma\mathrm{d}W_t)}
$$
where $W_t$ is a standard brownian motion.
When $\beta=0$, the whole thing continues to be a brownian motion (albeit with drift and with non-unit variance). In particular, there is a non-zero probability that $S_t$ will take on negative values. Conversely, when $\beta=1$ this describes a particular instance of geometric brownian motion and as a result, there process almost certainly avoids taking on negative values.
My question: for $0 < \beta < 1$ (note: strict inequalties), what statements can be made about the probability of the process taking on negative values? In particular, are negative values almost never achieved for whenever $\beta < 1$, or is there some critical $0 < \beta_c < 1$ such that processes with $\beta < \beta_c$ are able to take on negative values with non-zero probability while those with $\beta > \beta_c$ are not? Does the answer depend on the value of $\mu$? I'm most interested in the case where $\mu=0$ but left the drift term there to see if someone could offer insight about the more general setup.
Best Answer
Assuming that $\mu$ and $S_0$ are positive, the process stays almost surely non-negative. This is easily seen as when $S$ hits zero, it has a deterministic drift upwards. However, the process does not necessarily stay strictly positive, the hitting probability of zero can be positive.
This model is well studied in the financial mathematics literature and goes there under the name CEV model. The generic reference on it and the close relationship to Bessel processes is the following paper by Delbaen and Shirakawa: A Note of Option Pricing for Constant Elasticity of Variance Model, https://people.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf