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&=μX_tdt+σX^γ_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: $$ \lim\limits_{K\to\infty} \frac{1}{K^{2(1-γ)}}\log\mathsf{P}(τ_{0}\le T) =-\frac{1}{2\E M^2_T}, $$ where $M_T={\int }_0^T\sigma (1-\gamma )e^{-(1-\gamma )\mu s}d{B}_s$. Moreover we describe the most likely path to absorbtion of the normed process $\frac{X_t}{K}$ for $K\to \infty$.

10 pages

References in corpus (2)

• The Freidlin-Wentzell LDP with rapidly growing coefficients
• Likely path to extinction for simple branching model (Large Deviations approach)