ASYMPTOTIC ANALYSIS OF RUIN IN THE CONSTANT ELASTICITY OF VARIANCE MODEL

We give an asymptotic analysis for the probability of absorption P(tau(0) <= T) on the interval [0, T] of a nonnegative solution X-t of the following stochastic differential equation with respect to the Brownian motion B-t: dX(t) = mu X-t dt + sigma X-t(gamma) dB(t), X-0 = K > 0. tau(0) = inf{t : X-t = 0}, and the parameter gamma is an element of [1/2, 1) in the diffusion coefficient sigma x(gamma) assures P(tau(0) <= T) > 0. Our main result is lim(K ->infinity) 1/K-2(1-gamma) log P(tau(0) <= T) = -1/2E M-T(2), where M-t = integral(t)(0) sigma(1-gamma)e(-(1-gamma)mu s) dB(s). Besides we describe the most likely path to absorption of the normed process X-t/K for K -> infinity.