We give asymptotic analysis for probability of absorbtion $mathsf{P}(tau_0le T)$ 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 $sigma x^gamma$, $gammain [{1/2},1)$ is not Lipschitz continuous and assures $mathsf{P}(tau_0>T)>0$. Our main result: $$ limlimits_{Ktoinfty} frac{1}{K^{2(1-gamma)}}logmathsf{P}(tau_{0}le T) =-frac{1}{2E M^2_T}, $$ where $ M_T=int_0^Tsigma(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 $Ktoinfty$.
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