Asymptotic analysis of ruin in CEV model

Abstract

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 $\sigma x^\gamma$, $\gamma\in [{1/2},1)$ is not Lipschitz continuous and assures $\mathsf{P}(\tau_0>T)>0$. Our main result: $$ \lim\limits_{K\to\infty} \frac{1}{K^{2(1-\gamma)}}\log\mathsf{P}(\tau_{0}\le T) =-\frac{1}{2\E M^2_T}, $$ 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 $K\to\infty$.