Asymptotic analysis of ruin in CEV model

We give asymptotic analysis for probability of absorbtion $\mathsf{P}({\tau }_0\le T)$ on the interval $[0,T]$, where ${\tau }_0=\inf \{t:X_t=0\}$ and $X_t$ is a nonnegative diffusion process relative to Brownian motion $B_t$, dX_t&=\mu X_tdt+\sigma X^\gamma_tdB_t. X_0&=K>0 Diffusion parameter $\sigma x^{\gamma }$, $\gamma \in [1\mathrm{/}2,1)$ is not Lipschitz continuous and assures $\mathsf{P}({\tau }_0>T)>0$. Our main result: $$\underset{K\to \mathrm{\infty }}{\lim }\frac{1}{K^{2(1-\gamma )}}\log \mathsf{P}({\tau }_0\le T)=-\frac{1}{2\text{E}{M}_T^2},$$ where $ M_T=\int_0^T\sigma(1-\gamma)e^{-(1-\gamma)\mu s}dB_s $. Moreover we describe the most likely path to absorbtion of the normed process $\frac{X_t}{K}$ for $K\to \mathrm{\infty }$.