Ruin Analysis in the Constant Elasticity of Variance Model

We give results on the probability of absorption at zero of the diffusion process with non-Lipschitz diffusion coefficient
$$ dX_t=\mu X_tdt+\sigma X^\gamma_tdB_t, $$
with $X_0=K>>1$, and $1/2\le \gamma<1$. \quad In finance this is known as the Constant Elasticity of Variance Model and our results give information on the time to ruin $\tau_0=\inf\{t:X_t=0\}$. We show that $P(\tau_0\le T)>0$ for all $T$, give the probability of ultimate ruin, and establish asymptotics
$$ \lim\limits_{K\to\infty} \frac{1}{K^{2(1-\gamma)}}\log\mathsf{P}(\tau_{0 }\le T) =-\frac{1}{\sigma^2}\begin{cases} \frac{\mu}{(1-\gamma)[1-e^{-2\mu(1-\gamma)T}]} , & \mu\ne 0 \\ \frac{1}{2(1-\gamma)^2T}, & \mu=0. \end{cases} $$
In addition, an approximation to the most likely paths to ruin is given with the help of Freidlin-Wentzell's natural modification the LDP result.